•  ElKiiEY  A  "^  1^ 


LIBRARY 

unVemity  Of 


iDOaTxoK  Limu 


^ 


A 


0^ 


,V 


«• 


* 


WENTWORTH'S 
SERIES    OF    MATHEMATICS. 


First  Steps  In  Number. 

Primary  Arithmetic. 

Grammar  School  Arithmetic. 

High  School  Arithmetic. 

Exercises  in  Arithmetic. 

Shorter  Course  in  Algebra. 

Elements  of  Algebra.  Complete  Algebra. 

College  Algebra.  Exercises  in  Algebra. 

Plane  Geometry. 

Plane  and  Solid  Geometry. 

Exercises  in  Geometry. 

PI.  and  Sol.  Geometry  and  PI.  Trigonometry. 

Plane  Trigonometry  and  Tables. 

Plane  and  Spherical  Trigonometry. 

Surveying. 

PI.  and  Sph.  Trigonometry,  Surveying,  and  Tables. 

Trigonometry,  Surveying,  and  Navigation. 

Trigonometry  Formulas. 

Logarithmic  and  Trigonometric  Tables  (Seven). 

Log.  and  Trig.  Tables  (Complete  Edition). 

Analytic  Geometry. 


Special  Terms  and  Oircnlar  on  Applioation. 


HIGH   SCHOOL 
ARITHMETIC 

(Wentwortu  &  Hill's  Practical  Arithmetic). 


BY 

G.  A.  WENTWORTH,  A.M., 

POPE880R  OF  MATHEMATICS  IN  PHILLIPS  EXETER  ACADEMT. 


FOR  HIGH  SCHOOLS  AND  ACADEMIES. 


JOHFS^mLL 

Qdl  &  Mechanical  Eng^ieer. 

SAU  FRANCISCO,  CAL. 

BOSTON: 
PUBLISHED  BY  GINN  &  COMPANY. 

1888. 


Entered  accortiing  to  Act  of  CongresB,  in  the  year  1888,  by 

CI.  A.  Wkntwoutii, 
iu  the  oflice  of  the  Librarian  of  CougresB,  at  Washingion. 


Typoqiuphy  by  J.  8.  CusHiNO  k  Co.,  Bop-ism. 

PBKH8WOUK   UY   UiNN   &  Cu.,   Bu8TUN. 


GIFT 


\AJ48 


eouc 

LIBRARY 


PREFACE. 


THIS  edition  is  intended  for  teachers,  and  for  them  only.  The 
publishers  will  make  every  effort  to  keep  the  book  from 
pupils ;  and  teachers  are  urged  to  exercise  the  utmost  care  not  to 
lose  their  copies,  or  to  leave  them  where  pupils  can  have  access 
to  them. 

It  is  hoped  that  young  teachers  will  derive  great  advantage 
from  studying  the  systematic  arrangement  of  the  arithmetical  work, 
for  such  attention  has  been  paid  to  this  as  the  limitation  of  the  page 
would  allow. 

It  is  also  expected  that  many  teachers,  who  are  pressed  for 
time,  will  find  great  relief  by  not  being  obliged  to  work  out 
every  problem  in  the  Arithmetic. 

G.   A.  WENTWORTH. 
Phillips  Exetee  Academy, 
August,  1888. 


916 


AEITHMETIO, 


45.  Find  the  following  sums :  231  +  764 ;  341  +  57.8 ;  430.31  +  58.61 ; 
512.87  +  36.84  + 12.78  +  711.56  +  415.86. 


512.87 

36.84 

12.78 

231                   341. 

430.31 

711.56 

764                     57.8 

58.61 

415.86 

995                   398.8 

488.92 

1689.91 

46.   Add    1543.1  to  164.7; 

to  1728  ;    to  402.56  ; 

to  1897.3;    to 

475.34 ;  to  6897.65. 

164.7          1728.             402.56 

1897.3 

475.34          6897.65 

1543.1          1543.1          1543.1 

1543.1 

1543.1 

1543.1 

1707.8          3271.1         1945.66 

3440.4 

2018.44          8440.75 

47.  Add  1897.3  to  475.34 ; 

to  6897.65 ;   1 

bo  1728  ; 

to  402.56 ;  to 

164.7  ;  to  .5236  ;  to  2.71828. 

475.34              6897.65 

1728. 

402.56 

164.7 

1897.3                1897.3 

1897.3 

1897.3 
2299.86 

1897.3 

2372.64              8794.95 

3625.3 

2062. 

0.5236  2.71828 

1897.3  1897.3 


1897.8236 

1900.01828 

48.   Find 

the 

following  sums  : 

.7854  +  3.1416  +  2.71828  ;    .7854 

+  3.1416  +  30,103 

;  2.71828  +  402.56+1897.3;  2.7113  +  27.53  +  341.586. 

0.7854 

0.7854 

2.71828 

2.7113 

3.1416 

3.1416 

402.56 

27.53 

2.71828 

30,103. 

1897.3 

2284.57828 

341.586 

6.64528 

30.106.937 

371,8273 

ARITHMETIC. 


49.  Add  737.87  to  each  of  the  following  numbers :  111 
2304  ;  222  ;  263  ;  373  ;  262.13  ;  561.2  ;  32.35  ;  604.3. 


111. 

737.87 
848.87 


1011. 
737.87 
1748.87 


2304. 

737.87 
3041.87 


222. 

737.87 
959.87 


1011; 

263. 
737.87 
1000.87 


373. 

737.87 
1110.87 


262.13 

737.87 
1000. 


561.2 

737.87 
1299.07 


32.35 

737.87 
770.22 


604.3 

737.87 
1342.17 


50.  Find  the  five  sums  :  230.8  +  223  +  2.63  +  373.8  +  56.123  ; 
32.358  +  821.9  +  23.04  +  73.7 ;  202.3031  +  71.575  +  65.813  +  .0078 
+  7.377;  653.03  +  65.303  +  6.5033;  939.303  +  65.746  +  8.2794  +  681.28. 


230.8 
223. 

2.63 
373.8 
56.123 
886.353 


32.358 
821.9 

23.04 

73.7 
950.998 


202.3031 

71.575 

65.813 

0.0078 

7.377 

347.0759 


653.03 
65.303 
6.5033 

724.8363 


939.303 
65.746 
8.2794 
681.28 

1694.6084 


52. 


2.7182818 
3.1415927 
0.7853982 

6.6452727 


0.4342945 
4.8104774 
2.5399772 
7.7847491 


3.2808693 

2.5399772 

4.8104774 

10.6313239 


1.6093295 
15.4323487 
3.785 

20.8266782 


0.3047973 
0.3010300 
0.6213768 

1.2272041 


0.3937043 
0.3819660 
0.4342945 

1.2099648 


53. 


0.3010300 
0.6180340 
0.3819660 
1.30103 


0.6180340 
2.2360680 
1.7320508 
4.5861528 


0.3819660 
1.7320508 
1.4142136 
3.5282304 


2.2360680 

1.4142136 

15.4323487 

19.0826303 


15.4323487 
0.8450980 
0^ 

16.5414467 


3.785 

0.6213768 

1.6093295 

6.0157003 


TEACHERS     EDITION. 


0.6213768 

1.6093295 

0.3047973 

3.2808693 

3.2808693 

0.3937043 

0.3047973 

0.3937043 

2.5399772 

4.2070434 

5.2839031 

3.2384788 

0.3937043 

0.3047973 

2.7182818 

3.2808693 

0.4342945 

3.1415927 

4.8104774 

0.5235988 

0.5235988 

8.485051 

1.2626906 

6.3834733 

54.   2.7182818 

3.1415927 

0.7853982 

3.1415927 

0.5235988 

0.4342945 

0.7853982 

4.8104774 

4.8104774 

0.5235988 

2.5399772 

0.3937043 

0.4342945 

0.3937043 
11.4093504 

0.3047973 

7.603166 

6.7286717 

0.3937043 

1.6093295 

1.6093295 

3.2808693 

0.3047973 

0.6213768 

1.6093295 

0.3937043 

3.785 

0.4342945 

0.5235988 

0.264 

4.8104774 

0.4342945 

15.4323487 

10.528675 

3.2657244 

21.712055 

0.6213768 

3.785 

15.4323487 

0.264 

0.264 

1.4142136 

15.4323487 

15.4323487 

2.2360680 

0.3937043 

1.4142136 

0.3819660 

3.2808693 

1.7320508 
22.6276131 

0.6180340 

19.9922991 

20.0826303 

65.   0.4771213 

2.7182818 

0.3010300 

0.2908882 

3.1415927 

0.6180340 

1.6093295 

0.7853982 

0.3819660 

0.8450980 

0.4342945 

2.2360680 

0.3819660 

4.8104774 

1.7320508 

0.6180340 

2.5399772 

1.4142136 

0.3010300 

0.3937043 
14.8237261 

15.4323478 

4.523467 

22.1157111 

1 

AiMinmriiiu. 

1.6093295 

0.3937043 

3.785 

0.6213768 

0.3047973 

15.4323487 

3.785 

3.2808693 

0.6213768 

0.264 

1.6093295 

1.4142136 

15.4323478 

0.6213768 

3.2808693 

1.4142136 

3.785 

0.3047973 

1.7320508 

0.264 

4.8104774 

24.8583194 

10.2590772 

29.6490831 

57.  Add  by  doul 

ble  columns  -. 

45.68 

154.31 

73.86 

73.91 

296.85 

453.71 

78.54 

736.48 

137.64 

534.69 

345.19 

98.87 

134.70 

782.34 

643.48 

581.43 

78.43 

462.71 

1448.95 

2393.60 

1870.27 

498.50 

65.42 

621.65 

17.37 

638.34 

167.32 

684.29 

763.43 

856.96 

231.56 

809.31 

718.83 

210.10 

798.83 

501.49 

671.54 

835.78 

315.72 

643.53 

356.47 

768.44 

2956.89 

4267.58 

3950.41 

791.52 

32.54 

763.80 

504.83 

254.63 

78.23 

879.26 

63.27 

345.61 

243.97 

131.56 

26.73 

732.86 

506.72 

489.56 

47.95 

283.54 

812.36 

856.43 

345.83 

607.28 

497.65 

643.46 

219.07 

541.26 

708.91 

68.72 

616.72 

463.73 

216.78 

857.94 

67.74 

436.74 

6570.39 


3501.93 


4064.96 


TEACHERS     EDITION. 


69.  8-3-2=3;  (8 -3) -2  =  5-2  =  3;  8 -(3  -  2)  =  8-l  =  7. 
8-2-3  =  3;  (8-2)-3  =  6-3  =  3;  8-(2-3)  =  8-(-l)  =  9. 
18-(7-3)=18-4  =  14;  18  -  (3  -  7)  =  18 -(-4)  =  22  ; 
18  -  3  +  7  =  22. 

70.  The  following  questions  will  illustrate  the  meaning  of  minus 
numbers : 

Starting  90  miles  south  of  Chicago,  I  go  50  miles  due  north  ;  and 
the  next  day  80  miles,  still  north.    How  far  from  Chicago  am  I  now? 

-  90  +  50  +  80  =  -  90  +  130  =  40,  the  number  of  miles  north.  Ans. 

With  only  $67  I  undertake  to  pay  three  bills,  of  |47,  of  $13,  and 
of  $11.     Can  I  pay  the  bills  ?     How  much  shall  I  lack  ? 

67  -  (47  +  13  + 11)  =  67  -  71  =  -  4.     I  shall  lack  $  4.  Ans. 

73.   Subtract  123  from  each  of  the  numbers:    234,  343,  424,  555, 


234 

343 

424 

555 

676 

725 

123 

123 

123 

123 

123 

123 

111 

220 

301 

432 

553 

602 

839 

999 

1000 

10101 

5120 

123 

123 

123 

123 

123 

716  876  877  9978  4997 

74.  Subtract  456  from  each  of  the  numbers  :  789,  879,  978,  6378, 
6855,  6853,  7797,  7006,  3542,  4334,  9790,  3455. 

789  879  978  6378  6855  6853 

456  456  456  456  456  456 

333  423  522  5922  6399  6397 

7797  7006  3542  4334  9790  3455 

456  456  456  456  456  456 

7341  6550  3086  3878  9334  2999 

75.  What  is  the  difference  between  779  and  974  ?  368  and  249  ? 
479  and  2301  ?  2731  and  929  ?  708  and  394  ?  1 123  and  1072  ?  891 
and  773  ?  8103  and  5621  ?  19,001  and  3456  ?   792  and  2180? 


(5 

ARITHMETIC. 

974 

368 

2301 

2731 

708 

779 

249 

479 

929 

394 

195 

119 

1822 

1802 

314 

1123 

891 

8103 

19001 

2180 

1072 

773 

5621 

3456 

792 

51 

118 

2482 

15545 

1388 

76.  Subtract: 

$183.45 

$716.43 

$647.51 

$270.04 

76.47 

628.74 

549.64 

128.31 

$106.98 

$87.69 

$97.87 

$141.73 

$125. 

$247.93 

$641.87 

$56.27 

101.50 

129.47 

333.95 

29.89 

$23.50 


$118.46 


$307.92 


$26.38 


77.   Subtract  from  7854  each  of  the  numbers  :    788,  879,  567,  5006, 
6107,  578,  867,  894,  463,  4603. 


78. 


7854 
788 

7854 
879 

7854 
567 

7854 
5006 

7854 
6107 

7066 

6975 

7287 

2848 

1747 

7854 
578 

7276 

7854 

867 

6987 

7854 

894 

6960 

7854 
463 

7391 

7854 
4603 

3251 

3.1415927 
2.7182818 

0.7853982 
0.5235988 

4.8104774 
0.4342045 

0.4233109 

0.2617994 

4.3761829 

2.5399772 
0.3937043 

0.3937043 
0.3047973 

3.2808693 
0.3047973 

2.1462729 

0.088907 

2.976072 

3.2808693 
1.6093295 

3.785 
0.6213768 

15.4323487 
0.264 

1.6715398 

3.1636232 

15.1683487 

TEACHERS     EDITION. 


79. 


80. 


1.7320508 
1.4142136 

0.3178372 

2.2360680' 
1.7320508 

0.5040172 

3.1415927 
0.7853982 

2.3561945 
0.7853982 

1.5707963 
0.7853982 
0.7853981 


0.3819660 
0.6180340 


2.2360680 
0.3819660 

1.854102 

2.2360680 
0.6180340 

1.618034 

3.1415927 
0.5235988 

2.6179939 
0.5235988 

2.0943951 
0.5235988 
1.5707963 

1.4142136 
0.6180340 

0.7961796 


O.JioUolO 
0.3010300 

0.317004 

0.3819660 
0.3010300 

0.080936 


1.5707963 

0.5235988 

1.0471975 
0.5235988 
0.5235987 

0.6180340 
0.3819660 


1. 


81.  In  a  school  of  83  pupils,  37  are  girls ;  the  rest,  boys.     How 
many  boys  are  there  ?  83  —  37  =  46.  Ans. 

82.  Take  1787  from  21,205,  and  what  is  the  remainder  ? 

21,205-1787=19,418.  Ans. 

83.  Into  a  bowl  containing  338  fine  shot  I  poured  a  handful  more, 
and  the  bowl  then  contained  720.     How  many  did  I  pour  in  ? 

720-338  =  382.  Ans. 

84.  From  a  box  containing  209  oranges  I  took  a  basketful,  and 
left  163  oranges.     How  many  did  I  take  in  the  basket  ? 

209-163  =  46.  Ans. 

85.  The  minuend  being  1718.754,  and  the  subtrahend  1389.328, 
what  is  the  remainder  ?  1718.754  -  1389.328  =  329.426.  Ans. 

86.  If  the  minuend  was  6532.18,  and  the  remainder  1916.47,  what 
was  the  subtrahend  ?  6532.18  -  1916.47  =  4615.71.  Ans. 

87.  How  many  must  be  taken  from  729,434   in    order  to  leave 
613,488  ?  729,434  -  613,488  =  115,946.  Ans, 


ARITHMETIC. 


88.  How  many  must  be  taken  from  1,000,000  to  leave  817,259  ? 

1,000,000-817,259  =  182,741.  Ans. 

89.  Subtract  4187.94  from  8010.101. 

8010.101-4187.94  =  3822.161.  Ans. 

90.  Find  the  difference  between  8,765,420  and  9,873,210. 

9,873,210  -  8,765,420  =  1,107,790.  Ans. 

91.  In  a  till  are  if  391  in  bills,  $67.50  in  gold,  $39.75  in  silver, 
and  $2.77  in  copper  and  nickel.     How  much  money  is  in  the  till  ? 

$391  +  $67.50  +  $39.75  +  $2.77  =  $501.02.  Ans. 

92.  Starting  out  with  $315.75  in  one  wallet  and  $54.37  in  another, 

I  pay  the  grocer  $127.38;   the  butcher,  $64.17;    the  shoemaker, 

$21.40  ;  the  landlord,  $50;  the  tailor,  $35.     What  ought  I  to  have 

left? 

$127.38  $315.75 

64.17  54.37 

^^■"^^  $370.12 

^^-                                297.95 
35.  

— — — ;  $72.17  Ans. 

$297.95 

93.  On  a  bill  of  $753.43,  I  pay  $517.87.  How  much  do  I  still 
owe  ?  If  I  owe  $817.87,  and  have  but  $637.50,  how  much  do  I  lack 
of  being  able  to  pay  ? 

$753.43  $817.87 

517.87  637.50 


$235.56  $180.37 

94.  If  a  man  was  born  January  1,  1812,  how  old  was  he  January 
1,  1878  ?     How  old  December  31,  1857? 

1878     1     1  1857    12    31 

1812     1     1  1812       1       1 

66  45     11     30 

95.  America  wa.s  discovered  in  1492.     How  many  years  after  its 
discovery  was  each  of  the  following  events? 

Settlement  of  Florida,  1565  ;  of  Virginia,  1607 ;  of  Massachusetts, 
1620;  of  Quebec,  1608;  French  and  Indian  War,  1756;  Declaration 


TEACHERS     EDITION. 


of  Independence,  1776  ;  inauguration  of  Washington,  1789  ;  war  with 
England,  1812  ;  Mexican  War,  1846  ;  Civil  War,  1861. 

1565  1607  1620  1608  1756 

1492  1492  1492  1492  1492 

73  115  128  116  264 


1789 

1812 

1846 

1861 

1492 

1492 

1492 

1492 

284  297  320  354  369 

96.  How  many  days  in  common  years,  and  in  leap-years,  between 
January  1  and  March  1  ?  January  4  and  April  4  ?  February  5  and 
May  5  ?  February  7  and  October  7  ?  January  4  and  July  4  ? 
March  4  and  July  4  ? 


Between  January  1  and  March  1,  58  days 
Between  January  4  and  April  4,  89  days 
Between  February  5  and  May  5,  88  days 
Between  February  7  and  October  7,  241  days 
Between  January  4  and  July  4,  180  days 
Between  March        4  and  July       4,  121  days 


59  in  a  leap-year, 
90  in  a  leap-year. 
89  in  a  leap-year. 

242  in  a  leap-year. 

181  in  a  leap-year. 

121  in  a  leap-year. 


97.  The  sum  of  two  numbers  is  3  ;  their  difference,  1.  What  are 
the  numbers  ?  The  sum  of  two  numbers  is  5 ;  their  difference,  1. 
Eequired  the  numbers.  What  two  numbers  added  together  make  8, 
if  the  difference  of  the  numbers  is  2  ?  If  the  difference  is  0  ?  if  4  ? 
if6? 

(3-fl)^2  =  2)  (5-hl)-^2  =  3|  (8 +  2)  ^2  =  5 

(3  -  1)  --  2  =  1  r  (5  -  1)  -  2  =  2  )  (8  -  2)  --  2  =  3 

(8-f0)--2  =  4|  (8 +4)  ^2  =  6)  (8 +  6) -2  =  7 

(8  -  0)  -f-  2  =  4  i  (8  -  4)  -^  2  =  2  i  (8  -  6)  ^  2  =  1 

98.  If  the  minuend  is  9874,  and  remainder  3185,  what  is  the  sub- 
trahend ?  The  subtrahend  being  7659,  and  remainder  675.68,  what 
is  the  minuend  ? 

9874     minuend.  7659.         subtrahend. 

3185     remainder.  675.68     remainder. 


• 


6689    subtrahend.  8334.68     minuend. 


10  ARITHMETIC. 


99.  The  smaller  of  two  numbers  is  7.957.64328  ;  their  diflference  is 
.00087692.     What  is  the  larger  number  ? 

7.95764328  +  0.00087692  =  7.9585202.  Ans. 

100.  The  larger  of  two  numbers  is  7.95764328,  and  their  difference 
is  7.153485.     What  is  the  smaller  number  ? 

7.95764328  -  7.153485  =  0.80415828.  Am. 

101.  A  hired  man  pumps  out  of  my  cistern  in  one  hour  243.75 
gallons  ;  in  the  next  hour,  227.5  gallons  ;  in  45  minutes  more,  an 
additional  137.75  gallons ;  and  the  cistern  is  empty.  How  much  was 
in  it  ?  243.75  +  227.5  +  137.75  =  609.         ^  g^,^    ^^ 

102.  From  what  number  must  I  subtract  5  to  leave  7  ?  8  to  leave 
9?  From  what  number  must  I  subtract  5.1736  to  leave  8.1964? 
6.231  to  leave  9.6648  ?   74.213  to  leave  25.787  ? 

5  8  5.1736  6.231  74.213 

7  9  8.1964  9.6648  25.787 


12  17  13.37  15.8958  100. 

103.  What  must  be  subtracted  from  1  to  leave  .5  ?  to  leave  .53  ? 
to  leave  .532  ?  to  leave  .5236  ?  to  leave  .5235988  ? 

1.  1.  1.  1.  1. 

05  053  0532  05236  0.5235988 

05  047  0.468  04764  04764012 

104.  I  start  on  a  journey  of  3433  miles.  The  first  day  I  make 
428  miles ;  the  second  day,  511  miles ;  the  third,  497  miles  ;  the 
fourth,  513.  How  many  miles  of  my  journey  remained  for  me  at 
the  close  of  each  day  ?  How  many  miles  had  I  gone  at  the  close  of 
each  day  ? 


3433 

428 

3005  after  first  day. 

428  end  of  first  day. 

511 

511 

2494  after  second  day. 
497 

1997  after  third  day. 
513 

939  end  of  second  day. 
497 
1436  end  of  third  day. 
513 

1484  after  fourth  day.  1949  end  of  fourth  day. 


teachers'  edition.  11 

105.  Subtract  76,343  from  the  sum  of  61,932,  51,387,  5193,  4674, 
and  8199  ;  then  subtract  23,657  from  the  remainder. 

61,932  +  51,387  +  5193  +  4674  +  8199  =  131,385. 

131,385  -  76,343  =  55,042  ;  55,042  -  23,657  =  31,385.  Ans. 

106.  J.  bought  a  farm  and  stock  for  1 7633.90  ;  sold  off  the  stock 
for  1305.75  ;  then  sold  the  farm  for  $  7325.     What  did  he  lose  ? 

1 305.75  +  1 7325  =  1 7630.75  ; 
$7633.90 -1 7630.75  =  $3.15.  Ans. 

107.  If  I  gave  |4375  for  my  land,  and  paid  for  house,  barn,  sheds, 
and  fences,  1 2789. 50;  also  $973.75  for  horses,  cattle,  tools,  etc.; 
what  did  my  farm  and  stock  cost  ? 

If  I  sold  part  of  the  land  for  $675,  and  some  cattle,  etc.,  for  $  217.50, 
what  may  I  estimate  as  the  cost  of  what  I  have  left  ? 

$4375  +  $  2789.50  +  $973.75  =  $8138.25. 

$675 +  $217.50  =  $892.50. 

$  8138.25  -  $  892.50  =  $  7245.75.  Ans. 

108.  Alfred  the  Great  died  at  the  age  of  52,  a.d.  901.  In  what 
year  was  he  born  ?  William  the  Conqueror  began  to  reign  a.d.  1066, 
and  reigned  21  years.  In  what  year  did  he  die  ?  Socrates  was  born 
B.C.  469,  and  died  at  the  age  of  70.  In  what  year  did  he  die  ?  Plato 
was  born  B.C.  429,  and  died  at  the  age  of  82.  In  what  year  did  he 
die  ?  Demosthenes  died  at  the  age  of  60,  B.C.  322.  In  what  year  was 
he  born  ?  The  battle  of  Marathon  was  fought  B.C.  490  ;  560  years 
later  Jerusalem  was  destroyed  by  Titus.  In  what  year  was  Jeru- 
salem destroyed  ? 

901  1066  469  429  322  560 

52  21  70  82  60  490 

A.D.  849      A.D.  1087       B.C.  399      b.c.  347      b.c.  382      a.d.  70 

109.  John  has  158  cents,  James  has  271  cents  ;  James  gives  John 
56  cents.     Which  has  more  than  the  other,  and  how  many  more  ? 

158  271  215 

_56  56  214 

214  John's  cents.     215  James's  cents.         1  James's  excess. 


12 


ARITHMETIC. 


116.   Multiply  111  by  5 ;  123  by  3  ;  231  by  2 ;  114  by  3  ;  421  by 

4 ;  512  by  5  ;  4328  by  4  ;  1187  by  6  ;  1782  by  8  ;  8.287  by  7 ;  9.6198 
by  3  ;  62.818  by  7 ;  9.2758  by  8  ;  52.134  by  9. 

Ill  123  231  114  421 

5  3  2  3  4 

555  369  462  342  1684 


512 
5 

2560 

4328 
4 

17312 

1187 
6 

7122 

1782 
8 

14256 

8.287 
7 

58.009 

9.6198 
3 

62.818 
7 

9.2758 
8 

52.134 
9 

28.8594 


439.726 


74.2064 


469.206 


117.   Multiply  0.5235988  by  6;  0.7853982  by  4  ;  3.14159265  by  5, 
and  the  product  by  5. 

3.14159265 
5 


0.5235988 


3.1415928 


0.7853982 
4 

3.1415928 


15.70796325 
5 

78.53981625 


118.  Multiply  3.1416  by  11 ;  by  12  ;  by  10  and  by  3,  and  add  the 
two  results  ;  by  10  and  by  4,  and  add  the  results ;  by  9  and  by  6, 
and  add  the  results.  Multiply  2.236068  by  11 ;  by  6  and  by  7,  and 
add  the  results  ;  by  8  and  by  9,  and  add  the  results  ;  by  10  and  by 
7,  and  add  the  results  (compare  the  sum  of  these  two  products  with 
the  sum  of  the  last  two  product*) ;  by  10  and  by  8,  and  add  the 
results  ;  by  12  and  by  7,  and  add  the  results. 


3.1416 
11 

34.5576 


3.1416 
12 

37.6992 


3.1416 
10 

31.4160 


3.1416 
3 

9.4248 


31.4160 
9.4218 

40.8408 


3.1416 
10 

31.4160 


3.1416 
4 

12.5664 


31.4160 
12.5664 

43.9824 


3.1416 
9 

28.2744 


3.1416 
6 

18.8496 


TEACHERS     EDITION. 


13 


28.2744 
18.8496 

2.236068 
11 

2.236068 
6 

2.236068 

7 

15.652476 

13.416408 
15.652476 

47.1240 

24.596748 

13.416408 

29.068884 

2.236068 
8 

2.236068 
9 

17.888544 
20.124612 

2.236068 
10 

2.236068 

7 

17.888544 

20.124612 

38.013156 

22.360680 

15.652476 

22.360680 
15.652476 

2.236068 
10 

2.236068 
8 

22.360680 

17.888544 

40.249224 

2.236068 
12 

38.013156 

22.360680 

17.888544 

26.832816 

2.236068 
7 

26.832816 
15.652476 

15.652476        42.485292 

120.  How  much  is  10  times  3.14159265?  100  times?  a  million 
times?  What  will  10  barrels  of  apples  cost,  at  |3.75  a  barrel?  at 
$  2.17  ?  at  $  5.875  ?  How  much  will  100  barrels  cost  at  each  of  these 
prices,  and  at  $3,375  ?  at  ? 5.125? 

10  X  3.14159265  =  31.4159265  ;   100  X  3.14159265  =  314.159265 ; 
1,000,000  X  3.14159265  =  3,141,592.65  ;   10  x  $3.75  =  $37.50  ; 
10  X  $2.17  -  $21.70  ;   10  X  $5,875  =  $58.75  ;   100  X  $3.75  =  $375 ; 
100  X  $2.17  =  $217  ;   100  X  $5,875  =  $587.50  ; 
100  x$  3.375  =  $337.50;   100  x  $5,125  =  $512.50. 


122.  What  is  a  tenth  of  2.36  ?  a  hundredth  of  2.36  ?  a  thousandth 
of  0.63.  Write  the  second  members  of  the  following  equations,  and 
then  read  them : 


0.01  X  7.8      = 
0.1    X  0.065  = 


0.001  X  4.31 
0.01    X  0.012 


0.0001  X  23.31  = 


0.1  X  2.36  =  0.236  ;   0.01  x  2.36  =  0.0236  ;   0.001  x  0.63  =  0.00063  ; 
0.01  X  7.8  =  0.078 ;  0.001  x  4.31  =  0.00431 ;  0.0001  x  23.31  =  0.002331 
0.1  x  0.065  =  0.0065  ;  0.01  x  0.012  =  0.00012. 


14 


ARITHMETIC. 


123.  Find  the  cost  of  30  barrels  of  flour,  at  |3.27  a  barrel ;  of  70 
barrels,  at  $4.58;  of  90  barrels,  at  $6.76;  of  100  barrels,  at  |7.84; 
of  120  barrels,  at  1 8.57. 

BOX  $3.27  =  $98.10;  70 X  $4.58  =  $320.60 ;  90 X $6.76  =  $608.40; 
100  x$  7.84  =  $784;  120  x  $  8.57  =  $  1028.40. 


124.  Find  the  cost  of  0.03  of  a  barrel  of  oil,  at  $27,875  a  barrel ; 
of  0.7;  of  0.009;  of  0.17  ;  of  0.019;  of  0.13;  of  0.8  ;  of  0.83  ;  of 
0.014  of  a  barrel  ? 

0.03  X  $27,875  =  $0.83625  ;  0.7  X  $27,875  =  $19.5125  ; 
0.009  X  $27,875  =  $0.250875  ;  0.17  X  $27,875  =  $4.73875; 
0.019  X  $27,875  =  $0.529625  ;  0.13  x  $27,875  =  $3.62375  ; 
0.8  X  $27,875  =  $22.30  ;  0.83  X  $27,875  =  $23.13625; 
0.014  X  $27,875  =  $0.39025. 

126.   What  is  the  numerical  value  of  the  expressions  : 
30x8.75?  700x7.81?  300x7.85? 

0.07  X  6.975  ?  8000  x  65.432  ?         0.0009  x  10356.78  ? 

30  X  8.75  =  262.5  ;  700  x  7.81  -  5467  ;  300  x  7.85  =  2355  ; 
0.07  X  6.975  =  0.48825  ;  8000  x  65.432  =  523,456 ; 
0.0009  X  10356.78  =  9.321102. 


129.  Multiply  0.785398  by  each  of  the  following  numbers:  2;  20; 
3  ;  300 ;  5  ;  0.5  ;  0.005  ;  737  ;  7.37  ;  856  ;  85.6  ;  0.0856  ;  10  ;  1001 ; 
1.001 ;  954  ;  0.00954. 


0.785398 
2 

0.785398 
20 

0.785398 
3 

0.785398 

300 

1.570796 

0.785398 
5 

15.707960 

0.785398 
0.5 

0.3926990 
-5.78838326. 

2.356194 

0.785398 
0.005 

235.619400 

0.785398 
737 

3.926990 
0.785398  X  7.37  = 

0.003926990 

5497786 
2356194 
5497786 

578.838326 

TEACHERS     EDITION. 


15 


0.785398 
856 

4712388 
3926990 


0.785398  X  85.6  =  67.2300688. 
0.785398  X  0.0856  =  0.0672300688. 


6283184 

0.785398 
1001 

0.785398 

672.300688 

954 

785398 

3141592 

785398 

3926990 

786.183398 

7068582 
749.269692 

0.785398  X 

1.001  = 

0.786183398. 

0.785398  X  0.00954 

=  0.00749269692. 

0.785398 
10 

7.853980 


130.   Multiply  2150.42  by  0.1 ;    by  0.001 ;    by  0.75 ;   by  0.075 ; 
by  0.083. 


50.42 
0.1 

2150.42 
0.001 

2150.42 
0.75 

1075210 
1505294 

1612.8150 

2150.42 
0.075 

1075210 
1505294 

161.28150 

•  2150.42 
0.083 

5.042 

2.15042 

645126 

1720336 

178.48486 

131.  Multiply  1 .4142136  by  0.7 ;  by  0.707 ;  by  0.7071 ;  by  0.707107. 
Multiply  1.41421  by  1.4;  by  1.4142;  by  1.41422.  Multiply  1.732 
by  1.732 ;  2.23607  by  2.236 ;  0.618  by  618  ;  0.618034  by  0.618035. 
Subtract  this  last  product  from  1. 


1.4142136 

1.4142136 

1.4142136 

1.4142136 

0.7 

0.707 

0.7071 

0.707107 

0.98994952 

98994952 

14142136 

98994952 

98994952 

98994952 
98994952 

14142136 

0.9998490152 

98994952 

0.99999043656 

98994952 

1.0000003360552 

16 

AEITHMETIC. 

1.41421 

1.41421 

1.41421 

1.732 

1.4 

1.4142 

1.41422 

1.732 

565684 

282842 

282842 

3464 

141421 

565684 

282842 

5196 

1.979894 

141421 
565684 

565684 
141421 

12124 
1732 

141421 
1.999975782 

565684 
141421 

2.999824 

2.0000040662 

2.23607 

0.618 

0.618034 

1. 

2.236 

618 
4944 

0.618035 

0.38196664319 

1341642 

3090170 

0.61803335681 

670821 

618 

1854102 

447214 

3708 

4944272 

447214 
4.99985252 

381.924 

618034 
3708204 

0.381966643190 
133.   Find  the  value  of  the  expressions  :  88  X  718.54  ;  96  x  6.8193  ; 
6.3  X  71.569  ;  1.32  x  234.769. 

718.54  6.8193  71.569  234.769 

11  12  0.9  0.12 


7903.94 
8 

63231.52 


81.8316 


654.6528 


64.4121 

7 

450.8847 


28.17228 
11 

309.89508 


134.   Multiply  291.47  by  16,  and  the  product  by  625. 
In  like  manner,  find  the  continued  products  • 

8  X  125  X  278.56  ;  8  X  3.75  X  3.33333 ; 
8  X  625  X  1.5708. 

8  X  125  X  278.56  =  1000  X  278.56 

«=  278,560. 
8  X  3.75  X  3.33333  =  30  X  3.33333 
-  99.9999. 

8  X  625  X  1.5708  -  5000  x  1.5708 
-7864. 


291.47 
16. 

174882 
29147 

4663.52 
625 

2331760 
932704 
2798112 

2914700.00 


TEACHERS     EDITION. 


17 


135.    One  mile  measures  5280  feet.     How  many  feet  in  3  tenths  of 
a  mile  ?  in  0.7  ?  in  0.17  ?  in  0.573  ?  in  0.846  of  a  mile  ? 

0.3  X  5280  =  1584  ;  0.7  X  5280  =  3696  ;  0.17  X  5280  =  897.6  ; 
0.573  X  5280  =  3025.44  ;  0.846  x  5280  =  4466.88. 


138.  Multiply  (using  complements)  0.7854  by  9.9 ;  by  0.99  ;  by 
0.099.  Multiply  0.5236  by  99.7  ;  by  9.989  ;  by  9.87.  Multiply  8537 
by  0.0097  ;  by  0.9995. 


0.7854  X  10 

=  7.854 

0.7854  X  1 

=  0.7854 

0.7854  X  0.1 

=  0.07854 

0.7854  X  0.01 
0.7854  X  0.99 

=  0.007854 

0.7854  X  9.9 

=  7.77546 

=  0.777546 

0.7854  X  0.1 

=  0.07854 

0.5236  X  100 

=  52.36 

0.7854  X  0.001 

=  0.0007854 

0.5236  X  0.3 
0.5236  X  99.7 

=  0.15708 

0.7854  X  0.099 

=  0.0777546 

=  52.20292 

0.5236  X  10 

=  5.236 

0.5236  X  10 

=  5.236 

0.5236  X  0.11 

=  0.0057596 

0.5236x0.13 
0.5236  X  9.87 

=  0.068068 

0.5236  X  9.989 

=  5.2302404 

=  5.167932 

8537  XO.OI 

=  85.37 

8537  Xl 

=  8537 

8537  X  0.0003 

=  2.5611 
=  82.8089 

8537  X  0.0005 
8537  X  0.9995 

=   4.2685 

8537  X  0.0097 

.-=  8532.7315 

139.   Multiply  0.61803  by  147 ;  by  373 ;  by  7.56 ;   by  8.93  ;   by 
9.93.     Multiply  0.5236  by  5.99  ;  by  7.99  ;  by  8.997 ;  by  699.98. 


0.61803 

0.61803 

0.61803 

0.61803 

0.61803 

147 

373 
185409 

7.56 

8.93 

9.93 

432621 

370818 

185409 

185409 

247212 

432621 

309015 

556227 

556227 

61803 

185409 

432621 

494424 

556227 

90.85041 

230.52519 

4.6723068 

5.5190079 

6.1370379 

18 


ARITHMETia 


0.5236 
5.99 

47124 
47124 
26180 

3.136364 


0.5236 
7.99 

47124 
47124 
36652 

4.183564 


0.5236 
8.997 


699.98 
0.5236 


140.   Multiply  0.7854  by  0.618  ; 
Multiply  2.718  by  0.618  ;  by  0.382 

0.7854  0.7854 

0.618  0.382 


36652 
47124 
47124 
41888 

419988 
209994 
139996 
349990 

4.7108292      366.509528 

by  0.382  ;  by  0.7854  ;  by  0.302. 
by  0.7854  ;  by  0.607. 

0.7854         0.7854 
0.7854         0.302 

62832 
7854 
47124 

0.4853772 


2.718 
0.618 

21744 
2718 
16308 

1.679724 


15708 
62832 
23562 

0.3000228 


2.718 
0.382 

5436 
21744 
8154 

1.038276 


31416 
39270 
62832 
54978 

0.61685316 

2.718 
0.7854 

10872 
13590 
21744 
19026 

2.1347172 


15708 
23562 

0.2371908 


2.718 
0.607 

19026 
16308 

1.649826 


141.  Find  the  continued  products :  0.477  X  101  X  0.708  ;  15.43  X 
0.4343  X  3  ;  4  X  0.175  X  3.28 ;  0.615  x  0.771  X  10  ;  3.2809  x  5  x  0.71 ; 
0.785  X  0.7  X  0.202 ;  0.471  X  0.807  X  22  ;  3.28  x  25  x  0.909. 

15.43  0.175         0.615 


0.477 
101 


0.4343 


0.771 


477 

4629 

0.7^ 

615 

477 

6172 

3.28 

4305 

48.177 

4629 

2.296 

4305 

0.708 

6172 

0.474165 

385416 

6.701249 
3 

10 

337239 

4.74165 

34.109316 

20.103747 

TEACHERS     EDITION. 


.362134 


19 


3.2809 
5 

0.785 
0.7 

0.5495 
0.202 

0.471 

0.807 

3297 
3768 

0.380097 

22 

3.28 
25 

16.4045 
0.71 

1640 
656 

164045 
1148315 

10990 
10990 

82.PP 
0.909 

11.647195 

0.1109990 

760194 
760194 

738 
738 

74.538 


144.  Find  the  product,  to  the  fifth  fractional  place,  of  3.14159265 
by  2.236.  Find  1414.2136  x  M142.136,  to  the  second  place  ;  0.618034 
by  0.618034,  to  the  sixth  place  ;  2.236068  by  2236.068,  to  the  third 
place;  1.73205  by  1732.0508,  to  the  second  glace. 


3.14159265 

1414.2136000 

0.6180340 

6322 

63124141 

14142136000 

4308160 

6283185 

3708204 

628318 

5656854400 

61803 

94248 

141421360 

49442 

18849 

56568544 

2828427 

141421 

42426 

8485 

185 

7.024600 

24 

0.3819658 
0.381966.  Ans. 

20000001.063 

2.2360680 

1.732050 

8006322 

80502371 

44721360 

1732050 

4472136 

1212435 

670820 

51962 

134163 

3464 

1341 

87 

178 

1 

4999.9998 

2999.999 

5000.  Ans. 

3000.  Ans. 

20  ARITHMETIC. 


147.  1.  What  will  a  man  earn  in  a  year  if  he  has  $2  a  day,  omit- 
ting Sundays  ?  Suppose  that  the  year  begins  on  Sunday  ?  Suppose 
the  year  to  be  leap-year,  and  not  begin  on  Sunday  ?  Suppose  it  leap- 
year,  and  to  begin  on  Saturday  ? 

(365  -  52)  X  $2  =  $626 ;  (365  -  53)  x  $2  =  $624  ; 

(366  -  52)  X  $2  =  $628 ;  (366  -  53)  x  $2  =  $626. 

2.  If  a  field  of  corn  averages  2  ears  to  a  stalk,  how  many  ears  on 
673  stalks  ? 

673 

2  real  multiplicand. 

1346  1346  ears.  Ana. 

3.  At  27  bushels  an  acre,  how  much  wheat  to  the  square  mile  of 
640  acres,  deducting  47  acres  for  roads  and  waste  land  ? 

640 
_47 

593 
27  real  multiplicand. 

4151 
1186 

16011  16,011  bn.  Ana. 

4.  How  much  money  would  be  required  to  give  $  7000  to  each  of 
7568  men? 

7568 

7000  real  multiplicand. 

5297G000  $52,976,000.  Am. 

5.  In  a  certain  book  of  378  pages,  the  words  average  7  letters  to  a 
word,  and  10  words  to  a  line.  There  are,  on  an  average,  29  lines  to 
a  page.     How  many  letters  in  the  book  ? 

378 
29  real  multiplicand. 

3402 
756 

10962 

10  real  multiplicand. 

109620 

7  real  multiplicand. 
767340  767.340  letters.  Ans. 


teachers'  edition.  21 

6.  How  many  bushels  of  wheat  in  a  township  of  37  square  miles, 
if  we  deduct  47  acres  to  the  square  mile  for  roads  and  waste,  and 
suppose  that  half  the  remainder  is  in  wheat  averaging  23  bushels  to 
an  acre  ? 

47  640 

37  37 

329  4480 

141  1920 


1739  23080 

1739 


21941 
0.5 


10970.5 

23  real  multiplicand. 

329115 
219410 


252321.5 

252,321.5  bu.  Ans. 


7.  If  5700  persons,  each  paying  1  cent  toll,  and  324  carriages,  each 
paying  5  cents  toll,  pass  over  a  bridge  in  a  day,  how  much  money 
will  be  received  ? 

5700  324 

0.01  real  multiplicand.  0.05  real  multiplicand. 

57.00  16.20 

57 


73.20 


$73.20.  Ans. 


8.  A  merchant  bought  960  pounds  of  cheese  at  7  cents  a  pound, 
and  147  pounds  of  butter  at  20  cents.  He  gave  in  payment  12.5 
yards  of  cloth  at  1  dollar  a  yard,  2  barrels  of  sugar,  each  weighing 
226  pounds,  at  9  cents  a  pound,  and  the  remainder  in  cash.  How 
much  money  had  he  to  pay  ? 


22  ARITHMETIC. 


960  226 

0.07  real  multiplicand.  2  real  multiplicand. 

67.20  452 

0.09  real  multiplicand. 

40.68 
12.50 


147  53.18 

0.20  real  multiplicand. 

29.40  96.60 

67.20  53.18 

96.60  43.42  $43.42.  Ans. 

148.    1.   Express  the  product  of  7^  X  7^  8^  x  8  ;  2^  x  2 ;  5*  x  5^ 
75x73^78.   82x8  =  83;   2«x2  =  29;   5*x52  =  5«. 

2.  Express  the  product  of :  3.0Px  3.01 ;  0.672x  0.678 ;  0.208x0.208'. 

3.0P  X  3.01  =  3.013 ;   0.67^  x  0.678  =  0.67i«> ; 
0.208  X  0.208'  =  0.208*. 

3.  Express  the  product  of : 

2.0032x2.003*;    20.033x20.03;   20.03x20.032. 
2.0032  X  2.003*  =  2.003«  ;  20.03'  x  20.03  =  20.03* ; 
20.03  X  20.032  =  20.03'. 

153.   Divide  963  by  3  ;  846  by  2 ;  846  by  3  ;  846  by  6  ;  848  by  4  ; 
52.05  by  5  ;  84.028  by  7  ;  13.31  by  11  ;  1.728  by  12. 
3)963  2)846  *       3)846 

321  423  282 

6)846  4)848  5)52.05 

141  212  10.41 

7)84.028  11)13.31  12)1.728 

12.004  1.21  0.144 

158.   1.  Divide  0.003  by  0.07 ;  0.003  by  110  ;  110  by  0.003. 
7)0.30000  11)0.000300  SJUOOOOOOOOO 

0.04286  0.000027  36666.66667 

2.   Divide  0.07  by  0.003  ;  110  by  0.07  ;  1.3  by  0.07. 

3)  70.00000  7)11000.()00(X)  7)  130.00000 

23733333  1571.42857  18.57143 


teachers'  edition.  23 

3.  Divide  1.7  by  0.07  ;  0.07  by  110  ;  1.3  by  110. 

7)170.00000  11)0.00700  11)0.13000 

24.28571  0.00064  0.01182 

4.  Divide  1.7  by  110  ;  0.07  by  1.2  ;  0.003  by  1.2. 
11)0.17000  12)0.70000  12)0.0300 

0.01545  0.05833  0.0025 

5.  Divide  110  by  1.2  ;  1.7  by  1.2;  17  by  1.2. 
12)1100.'00000  12)17.00000  12)170.00000 

91.66667  1.41667  14.16667      " 

6.  Divide  136  by  0.06  ;  136  by  0.12  ;  136  by  1100. 

6)13600.00000        12)13600.00000  11)1.36000 

2266.66667  1133.33333  0.12364 

7.  Divide  256  by  0.8  ;  2.56  by  0.08  ;  0.0256  by  0.008. 

8)  2560  8)256  8)25.6 

320  32  3.2 

8.  Divide  256  by  8000  ;  1.06  by  0.9  ;  1.06  by  9000. 

8)0.256  9)10.60000  9)0.00106 

0.032  1.17778  0.00012 

160.    1.  Divide  1.6093295  by  0.479  ;  by  0.917  ;  by  0.017  ;  by  0.0087. 


479) 

3.35977 
1609.3295 
1437 

917) 

1.75499 
1609.3295 
917 

6923 
6419 

94.66644 
17)1609.3295 
153 

79 
68 

113 
102 

112 
102 

109 
102 

75 
68 

87) 

184.98040 
16093.2950 

87 

1723 
1437 

2862 
2395 

739 
696 

5042 

4585 

433 

348 

4679 
4311 

4579 
3668 

852 
783 

3685 
3353 

332 

9115 
8253 

862 

699 
696 

350 

34« 

24 

ARITHMETIC. 

2.   Divide  3  by  1.7;  by 

1.73; 

by 

1.732;  by  1.7321. 

1.76471 

1.73410 

17)30.0000 

173)  300.0000 

17 

173 

130 

1270 

119 

1211 

110 

590 

102 

519. 

80 

710 

68 

692 

120 

180 

119 

173 

1 

7 

1.73210 

173200 

1732)3000.0000 

17321)30000.0000 

1732 

17321 

12680 
12124 


5560 
5196 

3640 
3464 

1760 
1732 


126790 
121247 


55430 
51963 

34670 
34642 

280 


28 

3.  Divide  1.60932nr)  l,y  r.i>so,  and  the  quotient  by  12. 
0.00o;U)l.S  0.0O()0254 

528)0.16093295  12)0.0003048" 

1584 

2532 
2112 


4209 


TEACHERS     EDITION. 


25 


4.  Divide  2  by  1.4142  ;  5  by 
1.41423 

2.236. 

2236) 

2.23614 

14142)  20000.00000 
14142 

5000.0000 
4472 

58580 
56568 

5280 
4472 

8080 
6708 

13720 
13416 

20120 
14142 

59780 
56568 

32120 

28284 

3040 
2236 

3836 


804 


165.    Perform  the  work  in  the  following  questions  by  the  use  of 
reciprocals  : 

1.  8x0.25    =8h-4  =  2. 

2.  171 -H  0.25   =171x4  =  684. 


9.  567  ^  625  =  (567^  5)x  0.008 
=  113.4  X  0.008 
=  0.9072. 


3.  876x1.25   =876^0.8 

=  8760  -  8 
=  1095. 

4.  132x2.5     =132-^-0.4 

=  1320  ^  4 
=  330. 

5.  591  H-  2.5      =  591  x  0.4 

=  236.4. 

6.  756^0.125=756x8 

=  6048. 

7.  268x25      =268-0.04 

=  26800  -  4 
=  6700. 

8.  753-5-25      =753x0.04 

=  30.12. 


10.  1764x0.025  =  1764-40 

=  44.1. 

11.  5381^0.025  =  5381x40 

=  215,240. 

12.  7452  ^  0.875  =  7452  x  8  h-  7 

=  59,616-^7 
=  8516.6. 

13.  651    X  0.33333  =  651 -5- 3 

=  217. 

14.  456   X  6.66667  =  456  ^  0.15 

=  45,600  H-15 
=  3040. 

15.  1554x0.16667=  1554 -^  6 

=  259. 


26  ARITHMETIC. 


16.  432  +  1.33333  =  432x0.75  17.   375  +  16.667  =  375x0.06. 

=  324.  »  22.50. 

18.   225  +  6.6667  =  225  x  0.15 
=  33.75. 
167.    1.  Taking  7  as  unity,  what  would  be  the  value  of  14?  of  28? 
of  35?   of  3.5?   of  2.8105?   of  6.31415? 

1]U        7)28        7)35        7}3^        7)2.8105        7)6.31415 
2  4  5  0.5  0.4015  0.90202 

2.    If  the  side  of  a  square  is  10  inches,  and  its  diagonal  14.14214, 
express  the  side  in  terms  of  the  diagonal  as  unity. 

0.70710 


1414214)  1000000.00000 
9899498 

10050200 
9899498 


1507020 
1414214 


928060 

3,  If  the  diagonal  of  a  square  is  one  foot,  what  decimal  of  a  foot 
must  its  side  be  ?  i  ^  1.414214  =  0.70710.  (See  167.  2.) 

4.  If  the  diameter  of  a  circle  is  11.3  inches,  and  its  circumference 
35.5  inches,  what  is  the  circumference  in  terms  of  the  diameter  i* 
"What  is  the  diameter  in  terms  of  the  circumference  ? 

3.14159  0.31831 

113)355.00000  355)113.00000 

339  1065 

160  650 

113  355 

470  2950 

452  2840 

180  1100 

113  1065 

670  350 

S65 

1050 
1017 


teachers'  edition.  27 


5.   What  decimal  fraction  of  87  is 

47?  53?  43.5? 

29? 

0.54023 

0.60919 

0.5 
87)43.5 

0.333 

87)47.00000 

87)53.00000 

87)  29.000 

435 

622 

43.5 

261 

350 

800 

290 

348 

783 

261 

200 

170 

290 

174 

87 

261 

260 

830 
783 

29 

6,   How  many 

times  393  is  587  ? 

7857?   131?   196.5? 

1.49 

19.99 

0.33 

0.5 

393)587.00 

393)7857.00 

393)131.00 

393)  196.5 

393 

393 

1179 

196.5 

1940 

3927 

1310 

1572 

3537 

117? 

3680 

3900 

131 

3537 

3537 

363 

7.   How  many  684'8  are  there  in  1368  ?  in  1760  ?  in  342  ?  in  77  ? 
in  6.84?  in  0.0785? 

2  0.5  0.01 

684)1368  684)342.0  684)^84 

1368  342  0  6  84 


2.57 

0.1126 

684) 

0.00011 

684)  1760.00 
1368 

684)  77.000 

684 

860 

684 

1760 
1368 

0.0785 
684 

3920 
3420 

101 

5000 

4788 

392 


28 


ARITHMETIC. 


168.  1.  Divide  11.4285285 
by  3.1415927  to  six  decimal 
])laces. 

3.637813 

'6HXm^)  114285285 
94247781 

20037504 
18849556 


4.    Divide  0.0053  by  72.654  to 
eight  decimal  places. 

0.00007294 
72^^^)5.30000 
508578 

21422 
14531 


1187948 
942478 


6891 
6539 


245470 
219911 

by 

ices. 

5.   Divide  6 
decimal  places 

352 

300 

25559 

25132 

427 

by  0.1573  to  three 
38.144 

314 

113 

94 

2.    Divide     0.004239239 
3.2783278  to  five  decimal  pk 
0.00129 

160000.0 
4719 

12810 
12584 

2260 
1573 

32783^7^)42392.39 
3278328 

629 

960911 
655675 

6. 

eight 

Divide  0.11  by  1937.^ 
decimal  places. 

305236 
295049 

\nm) 

0.00005677 
11.0000 

3.  Divide  437  by  215.253  to 

three  decimal  places. 

2.030 

21^3)437000 
43051 

96872 

13128 

11624 

1504 

1356 

649 
646 

148 
133 

teachers"  edition.  29 


7.   Divide  46  by  0.00751515151  to  three  decimal  places. 

6120.968 


75;^;^;:^^)  4600000000000 
450909091 


9090909 
7515152 

1575757 
1503030 


72727 
67636 

5091 
4509 

582 

169.  1.  Find  the  value  of  100  ;  iqi  .  102.  iqs  .  iq*  ;  iQS  .  iqs. 
10»  =  1;  101  =  10;  102  =  10x10=100;  10^  =  10x10x10=1000; 
10*  =  10  X 10  X 10  X 10  =  10,000 ;  10^  =  10  X 10  xlO  xlO  xlO  =  100,000 ; 
10«  =  10  X 10  X  10x10x10x10  =1,000,000. 

2.  Find  the  value  of  lO^ ;  10-^ ;  IO-2 ;  10-^ ;  10"*  ;  IQ-^  ;  10-«. 
10»  =  1 ;  10-1  _  0.1 ;  10-2  =  0.12 _  0.01 ;  10-3  _  o.l^  =  0.001 ; 

10-*  =0.1*  =  0.0001;  10-5  =  0.15  =  0.00001;  10-6  =  0.16  =  0.000001. 

3.  Find  the  value  of  100  X  100  ;  lOixlQ-i;  102x10-2;  10^x10  '; 
10*  X  10-5. 

100  X  100  =  100  =  1  .  101  X  10-1  _  100  =  1  ;  102  X  10-2  _  100  =  1  ; 

103x10-3=100  =  1;  10*  X  10-5  =  10-1  =  0.1. 

4.  Find  the  value   of   10^  -  lO-i ;      IO-2  -  IO2 ;      lO^i  -5-  lO"-^ ; 
20-2  ^  10-*. 

103  -^  10-1  _  10*  =  10,000 ;  10-2  -V- 102  =  10-*  =  0.1*  =  0.0001 ; 
10-1  ^.  10-3  =  102  =  100 ;  10-2  -=- 10-*  =  102  _  100. 

5.  Findthe  value  of  10-3x102;  103-^102;  10-3-^102;  10-2-^-10-3. 
10-3  y  102  =  10-1  =  0.1  ;  103  ^  102  =  101  =-10 ; 

10  3  +  102  =  10-5  _  0.15  _  0.00001  ;  10-2-10-3  =  lOi  =  10. 


30  ARITHMETIC. 


6.  Find  the  value  of  102  +  10  ;  102 -h  10^ ;  10»^10  ^  lO'^  +  lO-^ 
10' +  10  =  10;  102 -^10'  =  10-1  =  0.1;  100^10-1  =  10; 

10-1  + 10-1  =  10»  =  1. 

7.  Find  the  value  of  l.OP-r- 1.01-1;  l.OPxl.Ol-i;  l.Ol-'-hl.Oii. 
l.OP  + 1.01-1  _  1.013  _  1.030301 ;  1.01«  X  1.01-1  =  1  01 ; 

1.01-2  ^  1.01-1  ^  1.01-1  =  1  -i- 1.01  =  0.99009. 


Exercise  I. 

1.  Express  in  words,  327.244. 

Three  hundred  twenty-seven  and   two  hundred  forty-four  thou- 
sandths. 

2.  Express  in  words,  80.9056. 

Eighty  and  nine  thousand  fifty-six  ten-thousandths. 

3.  Express  in  words,  0.390012. 

Three  hundred  ninety  thousand  twelve  millionths. 

4.  Express  in  words,  20000.002. 
Twenty  thousand  and  two  thousandths. 

6.  Express  in  words,  0.0000008. 
Eight  ten-millionths. 

6.  Express  in  words,  41.27105. 

Forty-one  and  twenty-seven  thousand  one  hundred  five  hundred- 
thousandths. 

7.  Write  in  figures,  two  hundred  thirty-five  and  eight  hundred 
thirty -five  thousandths. 

235.835. 

8.  Write  in  figures,  seventy-four  and  two  hundred  three  thou.-Jaiid 

six  millionths. 

74.203006. 

9.  Write  in  figures,  twelve  hundred  and  eight  thousand  three  tjn- 

millionths. 

1200.0008003. 


teachers'  edition.  31 

10.  Write  in  figures,  five  thousand  sixty-four  millionths. 

0.005064. 

11.  Write  in  figures,  one  million  and  four  tenths. 

1000000.4. 

12.  Write  in  figures,  six  hundred-millionths. 

0.00000006. 

13.  Multiply  and  divide  789.365  by  10  ;  by  100  ;  by  100,000. 
7893.65;  78.9365;  78936.5;  7.89365;  78,936,500;  0.00789365. 

14.  Multiply  and  divide  0.004  by  100 ;  by  10,000 ;  by  1000. 
0.4  ;  0.00004 ;  40  ;  0.0000004 ;  4  ;  0.000004. 

15.  Multiply  and  divide  436  by  1,000,000  ;  by  1000  ;  by  10. 
436,000,000;  0.000436;  436,000;  0.436;  4360;  43.6. 

16.  Multiply  and  divide  0.1  by  10  ;  by  ten  millions. 
1;  0.01;  1,000,000;  0.00000001. 

17.  Find  the  value  of  21.3706  +  15.243  +  1.8954  +  0.26891  +  5.328 
H-  29.74. 

21.3706 
15.243 

1.8954 

0.026891 

5.328 
29.74 


73.603891 


18.  Find  the  value  of  57  +  0.0057  +  6.8  + 1200  +  0.847  + 159.2  +  3. 

57. 

0.0057 
6.8 
1200. 

0.847 
159.2 
3. 


1426.8527 


32  ARITHMETIC. 


19.  Find  the  value  of  0.0012  + 10  +  5.8281  +  5  +  39.43  +  0.6827  + 1. 

0.0012 
10. 

5.8281 

5. 
39.43 

0.6827 

1. 


61.942 
20.   Find  the  value  of  23.9875  -  12.4764  ;  35.14732  -  27.62815. 


23.9875 

35.14732 

12.4764 

27.62815 

11.5111 

7.51917 

21.  Find  the  value  of  102.1274  - 

-83.072;  39.801- 

102.1274 

39.801 

83.072 

17.9645 

17.9645. 


19.0554  21.8365 

22.   Find  the  value  of  30  -  5.2817  ;  1.7  -  0.8469. 

30.0000  1.7000 

5.2817  0.8469 


24.7183  0.8531 

23.   Find  the  value  of  1  -  0.54237  ;  100-0.00176. 

1.00000  100.00000 

0.54237  0.00176 


0.45763 

99.99824 

ind  the 

value  of  24.271  -  3.5485  + 15.271  - 

-13.256- 

3.6485 

24.271 

13.256 

39.542 

15.271 

14.125 

31.0295 

14.125. 


39.542  31.0295  8.5125 


TEACHERS     EDITION. 


33 


)5.   Fmd  the  value  of  52  +  0.52  -  17.8946  -  30.254  -  0.5  +  21.12. 
52.  17.8946 

0.52  30.254  73.64 

21.12  0.5  48.6486 


73.64 


48.6486 


26.   Find  the  value  of  41.289  x  0.5  ;  0.268  x  0. 
41.289  0.268 

0.5  0.9 


24.9914 

0.112  X  0.2. 
0.112 

0.2 


20.6445 


0.2412 


0.0224 


27.   Find  the  value  of  2.435  x  4.23  ;  71.651  x  3.37  ;  0.251  x  0.04. 


28 

0.0768. 


2.435 

71.651 

0.251 

4.23 

3.37 
501557 

0.04 

7305 

0.01004 

4870 

214953 

9740 

214953 

10.30005 

241.46387 

Find  the  value  of  0.0012  x  0.005  ; 

2.26823  x  200 ; 

0.0012 

2.26823 

5.6125 

0.005 

200 

0.0768 

0.000006 

453.646 

449000 
336750 
392875 

29. 


0.43104 
Find  the  value  of  0.7  X  7  X  0.07 ;  0.15625  x  23.7  X  0.00192  x  5. 


0.7 
7 

0.15625 
23.7 

109375 
46875 
31250 

3.703125 
0.00192 

4.9 

0.07 

0.343 

7406250 
33328125 
3703125 

3.703125 

0.00711 
5 

0.03555 


30.  Find  the  value  of  (2.465  +  1.121)  x 

(3.2- 

-  2.89). 

(2.465 +  1.21)  X  (3.2- 

-2.89; 

) 

-  3.675  X  0.31 

=  1.13925. 

31.   Find  the  value  of  (3.01)» ;  (0.045)' ;  (0.0081)» ;  (5.1004)» ; 

(0.76)». 

3.01                             0.045 

0.0081 

3.01                             0.045 

0.0081 

301                               225 

81 

903                                 180 

648 

9.0601                       0.002025 

0.00006561 

S.1004 

0.76 

ft.  1004 

0.76 
456 

204016 

51004 

532 

255020 

26.01408016 
5.1004 

10405632064 
2601408016 
13007040080 

132.682214448064 


0.5776 
0.76 

34656 
40432 

0.438976 


32.   Find  the  value  of  (0.125)«  x  (0.32)». 


0.125 
0.125 

625 
250 
125 


0.015625 


0.32 
0.32 

64 
96 


0.1024 
0.32 

2048 
3072 

0.032768 


0.032768 
0.015625 

163840 
a5536 
196608 

163840 

32708 


0.000512 


teachers'  edition.  35 


33.   Divide  291.84  by  6  ;  0.12936  by  12  ;  7.92801  by  0.9. 

6)291.84  12)0.12936  9)7.92801 

48.64  0.01078  0.88089 


34.  Divide  58.383  by  0.39  ;  0.28744  by  0.08  ;  491.205  by  0.065. 
149.7  7557 


))  5838.3 

8)28.744 

65)491205 

39 

3.593 

455 

193 

362 

156 

325 

378 

370 

351 

325 

273 

455 

273 

455 

35.  Divide  68.325  by  6.25  ;  0.732  by  1.6  ;  1208.88  by  0.438. 


10.932 

16j 

0.4575 

438) 

2760 

625)6832.500 

7.3200 

1208880 

625 

64 
92 

/ 

876 

5825 

3328 

5625 

80 
120 

3066 

2000 

2628 

1875 

112 

2628 

1250 

80 

1250 

80 

36.  Divide  498  by  0.0125;  7  by  0.007  ;  1000  by  0.0001. 

The  reciprocal  of  0.0125  is  80. 

498                            7)7000  1)10000000 

80                              1000  10000000 


39840 


36  ARITHMETIC. 


37.   Divide  0.235  by  10.24  ;  27  by  12  ;  0.00507702  by  0.0283. 
0.02295  0.1794 

1024)  23.;-;0000  12)27.00  283)50.7702 

2048  2.25  283 

3020  2247 

2048  1981 


9720  2660 

9216  2547 


5040  1132 

1132 


38.  Divide  89.3 

by  0.00752  ;  74.1  by  0.0256  ; 

1  by  0.128. 

11875 

2894.53125 

7.8125 

752) 8930000 

256)  741000.00000 

128)  1000.0000 

752 

512 

896 

1410 

2290 

1040 

752 

2048 

1024 

6580 

2420 

160 

6016 

2304 

128 

5640 

1160  » 

320 

5264 

1024 

256 

3760 

1360 

640 

3760 

1280 

640 

800 
768 

320 
256 

640 
512 

1280 
1280 

39.   Divide  0.39842  by  3.7164  ;  281.5  by  13.789  ;  0.0005  by  0.0028. 


TEACHERS     EDITION. 


37 


0.10720 

20.41482 

0.r78«7 

37164)  3984.20000 
37164 

13789} 

281500.00000 
27578 

28) 

1 5.00000 
28 

267800 
260148 

57200 
55156 

220 
196 

76520 
74328 

20440 
13789 

240 
224 

21920 

66510 
55156 

113540 
110312 

160 
140 

200 
196 

32280 

27578 

40.  Divide  63.04 
by  0.0059. 

0.06905 

128  by 
493) 

912.85;  287.209 

58257.40365 
28720900.00000 
2465 

by 
59; 

0.00493;  2000 

338983.05084 

91285)6304.12800 
547710 

)  20000000.00000 
177 

827028 
821565 

546300 
456425 

4070 
3944 

1269 

986 

230 
177 

530 
472 

2830 
2465 

580 
531 

3650 
3451 

490 

472 

1990 

1972 

180 
177 

1800 
1479 

300 
295 

3210 

2958 

500 
472 

2520 
2465 

280 
236 

38  ARITHMETIC. 


Exercise  II. 

1.   1.4  +  2.08  +  3.895  -  3.    1.667  +  0.4  +  0.286  + 

j^  +0.636+0.931  = 

2*08  1-G67 

3.895  OA 


7.375 


0.286 

6.08 

0.636 

0.931 


2.   2.8  +  2.08  +  0.28  +  0.028 

+  0.812  = 

2.8  10. 

2.08 

0  28  4.  6.125-0.57  = 

0.028  6.125 

0.812  0.57 

6.  5.555 

5.  (4.625 +  1.146) -(1.2 +  3.57)       6.  6.913  -  (2.85  -  0.937) 
=  5.771-4.771  =6.913-1.913 

=  1.  =5. 

7.  24  -  2.4  +  (5  -  3.508)  -  3.092 

=  24  -  2.4  +  1.492  -  3.092 
=  25.492  -  5.49^ 
=  20. 

8.  10  -  (4.25  -  2.5  +  2  -  0.625  -  0.4  -  2.02)  -  0.295 

=  10  -  (6.25  -  5.545)  -  0.295 
=  10  -  0.705  -  0.295 
=  10-1 
=  9. 

9.    1.5x0.08x0.5=  10    0.1204  xO:0168x  100 


1.5 
0.08 

0.12 
0.5 


0.06 


0.1204 
0.0168 

9632 
7224 
1204 

0.00202272 
100 

0.202272 


teachers'  edition.  39 


11.  0.04x3.25x0.06  = 

12.  36x0.00 

2x2.05x0.00765  = 

3.25 

36 

0.1476 

0.04 

0.002 

0.00765 

0.13 

0.072 

7380 

0.06 

2.05 

360 
144 

8856 

0.0078 

10332 

0.00112904 

0.1476 

13.  0.139x28+42x0.002 

+  6  X  0.004  -  0.05  X 

20 

=  3.892  +  0.084  +  0.024 

-1 

=  4-1 

=  3. 

14.  (10  - 1.25)  X  0.2  +  0.02  X  2.8  +  (80.3  X  0.1- 5.3)  X 10 -805.3x0.02 

=  8.75  X  0.2  +  0.02  X  2.8  +  (8.03  -  5.3)  X  10  -  805.3  X  0.02 
=  1.75  +  0.056  +  27.3  -  16.106 
=  29.106  - 16.106 
=  13. 

15.  28.3696-5-1.49=  .  17.  8.8779-^175.8  = 

19.04  ,  0.0505 


149)2836.96  1758)88.7790 

149  8790 


1346  8790 

1341  8790 


596 
596 


16.  0.27-5-0.00225=  18.  0.0427 -^  92.3  = 

120  0.00046 

225)27000  923)0.42700 

225  3692 

450  5780 

450  5538 


40 

ARITHMETIC. 

19. 

0.28744  ^  800  = 

8)0.0028744 

0.0003593 

20.  491.205^650  = 

22.   0.732-^16,000  = 

0.7557 

0.00001575 

65)49.1205 

16)0.00073200 

455 

04 

362 

02 

325 

80 

370 

120 

325 

112 

455 

80 

455 

80 

21.   68.325^6250  = 

23.    1208.88-0.438  = 

0.010932 

2760 

625)  6.832500 

438)1208880 

625 

876 

5825 

3328 

5625 

3066 

2000 

2628 

1875 

2028 

1250 

1250 

24.    2  ^  0.01-(0.2  -^  0.02  +  0.8  -^-10)  +  36.48  ^  8  -  (4  ^  0.05-2+0.0^1 .25 
=  200  -  (10  +  0.08)  +  4.56  -  (80  -  2  +  0.48) 
=  200  -  10.08  +  4.56  -  78.48 
«  204.56  -  88.56 
- 116. 


25.   72.2  -H  10  -  2  -h  (0.5  ^  1.60)  +  2.125  -!-  (1.75  -  0.5) 
=  72.2^10-2-^0.3125  +  2.125^1.25    . 
=  7.22-6.4  +  1.7 
=  8.92  -  6.4 


teachers'  edition.  41 


Exercise  III. 

1.   What  number  subtracted  88  times  from  80,005  will  leave  13 
as  a  remainder  ? 

909 


80,005  88)  79992 

13  792 


79,992  792 

792 

2.  If  7  men  can  build  a  wall  in  16  days,  how  many  men  will  it 
take  to  build  a  wall  three  times  as  long  in  half  the  time  ? 

7 
_3 

21 

42 

3.  How  many  minutes  are  there  between  25  minutes  past  8  in  the 
morning  and  midnight  ? 

35 
180 
720 

935 

4.  The  velocity  of  sound  being  1090  feet  per  second,  at  what  dis- 
tance is  a  gun  fired,  the  report  of  which  I  hear  11  seconds  after  seeing 
the  flash?     (5280  feet  make  a  mile.) 

2.27083 


1090 
11 

1090 
1090 

5280) 

11990.00000 
10560 

14300 
10560 

11990 

37400 
36960 

44000 
42240 

17600 
15840 

42  ARITHMETIC. 


5.    How  long  would  it  take  to  travel  30.2375  miles  at  the  rate  of 
8.85  miles  per  hour  ? 

3.4167 


885)  3023.7500 
2655 


3687 
3540 


1475 
885 

5900 
5310 

5900 
6195 

6.  The  circumference  of  a  circle  being  3.1416  times  the  diameter, 

find  the  circumference  of  a  circle  whose  diameter  is  6.8  feet ;  also,  find 
the  diameter  of  a  circle  whose  circumference  is  20  inches. 

6.366 
3.1416  31416)200000.000 

6.8  188496 


251328  115040 

188496  94248 


21.36288  207920 

-  21.363  ft.  Ans.  188496 


194240 

188496 


7.    How  much  wire  will  be  required  to  make  a  hoop  30  inches  in 
diameter,  allowing  two  inches  for  the  joining? 

3.1416 
30 


84.248 
_2 

96.248 


TEACHERS     EDITION. 


43 


8.   How  many  times  would  such  a  hoop  turn  in  going  half  a  mile? 

336. 


2)5280 
2640 

2640 
12 

5280 
2640 

31680 


94248)31680000. 
282744 


340560 
282744 

578160 
565488 


9.  Cork,  whose  weight  is  0.24  of  that  of  water,  weighs  15  pounds 
per  cubic  foot.  What  is  the  weight  of  6  cubic  feet  of  oak,  the  weight 
of  oak  being  0.934  of  that  of  water  ? 

62.5 
24)  1500.0  62.5 

144  6 


60 
48 

120 
120 


375. 
0.934 

1500 
1125 
3375 

350.25 


10.  From  what  number  can 
847  be  subtracted  307  times,  and 
leave  a  remainder  of  49  ? 

847 
307 

5929 
2541 

260029 
49 

260078 


11.   What  is  the  235th  part  of 
141,235? 

601 


235)141235 
1410 

235 
235 


44 


ARITHMETIC. 


12.    What  will  343  barrels  of 
flour  eost,  at  $6.37  a  barrel  ? 
$6.37 
343 

1911 

2548 
1911 


$2184.91 


13.  12  make  a  dozen,  and  12 
dozen  make  a  gross.  How  many 
steel  pens  in  28  gross  ?  What 
will  a  gross  of  eggs  cost,  at  27 
cents  a  dozen  ? 

144  $0.27 

28  12 


1152 

288 

4032 


54 

27 


$3.24 


14.  How  much  must  be  added 
to  $4429  in  order  to  make  the 
sum  43  X  $241? 

$241 
43 

723 

964 


10363 
4429 

$5934 


15.     What 

number  deducted 

from  the 

26th 

part  of  2262  wi  1 

leave  the 

87th 

part  of  the  same 

number  ? 

87 

26 

26)  2262 

87) 2262            87 

208 

174             26 

182 

522            61 

182 

522 

16.  At  an  ordinary  rate,  123  words  a  minute,  how  long  will  it 
take  a  man  to  deliver  a  speech  of  15  pages,  each  of  28  lines,  and  each 
line  containing  11  words?  How  long  would  it  have  taken  Daniel 
Webster  to  deliver  the  same  speech,  at  the  rate  of  93  words  a  minute? 

37.6  49.7 

123)4620.0 


15 

28 

120 
30 

420 
11 

420 
420 


369 

930 
861 

690 


93)4620.0 
372 

900 
837 

630 


4620 


TEACHERS     EDITION. 


45 


17.  How  long  would  it  take  a 

railway  train 

to   go  from  New 

York    to    San 

Francisco,    3310 

miles,  at  the  rate  of  1973  feet  a 

minute  ? 

8858 

3310 

1973) 17476800 

5280 

15784 

L'64S00 

16928 

6620 

15784 

inr)50 

11440 

17476800 

9865 

July  4,  how   much   money  will 
there  be  in  the  box  ? 


15750 


18.  How  long  will  it  take  to 
count  a  million,  at  the  rate  of  67 
a  minute  ? 

14925.4 


67)  1000000.0 
67 

330 

268 

620 
603 

170 
134 

360 
335 


250 


19.  If  you  put  into  a  box  17 
cents  a  day  including  Sundays, 
beginning  January  1  and  ending 


31 

185 

28 

0.17 

31 
30 
31 
30 

1295 

185 

31.45 

4 

.     =$31.45 

185 


20.  If  a  man's  income  is  1 3000 
a  year,  and  his  daily  expenses 
average  1 7.68,  what  does  he  save 
in  a  year  ? 


$7.68 
365 

3840 
4608 
2304 

12803.20 


13000. 
2803.20 

$196.80 


21.    In 

a  question  of  division 

the  quotient  was  87.83,  the  divi- 

sor 759. 

What   was   the   divi- 

dend? 

87.83 

759 

79047 

43915 

61481 

66662.9- 


46 


ARITHMETIC. 


22.  It  is  3.1416  times  as  far  round  a  wheel  as  across  it.  How 
many  times  will  a  wheel  4.5  feet  across  turn  round  in  going  23  miles 
of  5280  feet  each  ? 


8590 


5280 
23 

3.1416 
4.5 

141372^ 

1214400000 
1130976 

15840 
10560 

121440 

157080 
125664 

14.1372 

834240 
706860 

1273800 
1272348 

14520 


23.   How  many  gallons  of  231  cubic  inches  are  contained  in  a 

cubic  foot  (1728  cubic  inches)?  in  a  bushel  of  2150.42  cubic  inches? 
ITow  many  cubic  feet  in  a  bushel?  How  many  bushels  in  31.5 
gallons  ? 

(i.)  (ii.) 

7.48  9.309 


231)1728.00 
1617 

231, 

)  2150.420 
2079 

1110 
924 

714 
693 

1860 
1848 

2120 
2079 

(iii.) 

(iv.) 

1.244 

31.5 
231 

21504; 

3.38 

.728)2150.420 

1728 

2)  727650.00 
645126 

4224 
3456 

315 
945 
630 

7276.5 

825240 
645126 

7682 
6912 

1801140 
1720336 

TEACHERS     EDITION. 


47 


24.  Seven  children  had  left  to 
them  $7186  apiece;  one  died, 
and  his  share  was  divided  among 
the  surviving  six.  How  much 
had  each  then  ? 

6)17186.00 
$1197.67 
7186 


$8383.67 


26.  What  is  the  nearest  num- 
ber to  7196  that  will  contain  372 
without  a  remainder  ? 


19 


372)7196 
372 

3476 
3348 


7196 
128 

7068 


128 

26.  How  long  will  it  take  2 
men  to  do  what  1  man  can  do  in 
6  days  ?  what  4  men  can  do  in  3 
days  ?  what  3  men  can  do  in 
4  days  ? 

6  days  -7-2  =  3  days. 
2x3  days  =  6  days. 
(3x4  days)  -i-  2  =  6  days. 

27.  Divide  $1.80  am.ag  Thom- 
as, Richard,  and  Henry  in  such 
a  way  that  Henry  shall  receive 
3  cents  for  every  5  cents  that 
Thomas  gets,  and  Richard  shall 
receive  2  cents  for  every  3  cents 
that  Henry  gets. 


2 
3 
5 

10 

$0.18 
3 


10)  $1.80 

$0.18 

2 


$0.36  =  R.'8. 

$0.18 
5 


$0.54  =  H.'s.     $0.90  =  T.'s. 

28.   Divide  $87.84  between  B 

and  C  so  that  C  shall  get  $19  as 

often  as  B  gets  $17. 

2.44 

36)  87^84 


19 
17 


72 


36 

158 

144 

144 

144 

$2.44 

$2.44 

19 

17 

2196 

1708 

244 

244 

$46.36  =  B's. 

$41.48  = 

29.  Three  partners  received  for 
goods  :  one,  $371.63  ;  the  second, 
$285.40;  the  third,  $411.91. 
They  paid  for  the  goods  $879.34, 
and  divided  the  balance  equally 
among  them.  How  much  did 
each  receive  ? 
$371.63 


285.40 

411.91 

$1068.94 


$1068.94 

879.34 

3)  $189.60 

$   63.20 


48 


ARiTHMETIC. 


30.  At  12  inches  in  a  foot, 
how  many  inches  long  is  a  wall 
35  feet  in  length  ?  A  brick  and 
its  share  of  mortar  being  8.4 
inches  long,  how  many  bricks  in 
length  is  the  wall '' 


35 

ftO 

12 

84)4200 

70 

420 

35 

0 

420 

31,  A  brick  and  mortar  being 
2.4  inches  in  height,  how  many 
bricks  are  required  to  build  the 
wall  12  feet  high,  if  the  wall  be 
two  bricks  wide? 

12                  60  60 

12        24)U40  _^ 

144             ]^  3000 

0  2 

6000 

32.  What  is  the  total  weight 
of  the  wall,  if  a  brick  and  its  share 
of  the  mortar  weigh  4.13  pounds  ? 
What  is  the  weight  after  a  long 
rain,  when  the  weight  is  increased 
to  1.27  pounds  for  r>acli  brick'* 


4.13 
HOOO 

24780 


6000 


2r)(^)20 


33.    llow  many  pounds  does 

carli   foot  in  Vnt^th  of  tlio  wall 
weigh  ? 


708 
35)24780 
245 


280 
280 


35)25620 
245 
~112 
105 

'Id 

70 


34.  If  60.98  cubic  inches  of 
brick  weigh  4  pounds,  how  many 
cubic  inches  of  brick  weigh  1 
pound  ?  How  many  pounds 
would  a  cubic  foot  (1728  cubic 
inches)  weigh  ? 

4^60.980 


15.246 


113.35 

15245)  1 72801  lOoo 
io24r. 


20350 
15215 


5ia50 
46735 

~53l50 
45735 
74160 

35.  If  a  cubic  foot  of  wat«M- 
weigh  62.5  pounds,  how  many 
times  as  heavy  as  water  is  brick' 

1.8 


625)1133.5 
625 
5085 
5000 


TEACHERS     EDITION. 


49 


36.  Light  moves  through  the 
air  at  186,500  miles  in  a  second. 
How  many  times  can  it  go  around 
the  earth  in  a  second,  the  distance 
round  the  earth  being  24,897.714 

miles  ? 

7^ 

24897714)186500000.0 
174283998 

122160020 

37.  Light  moves  through  the 
air  at  300,190  kilometers  in  a 
second.  How  many  times  can  it 
go  around  the  earth  in  a  second, 
the  distance  round  the  earth  be- 
ing 40,007.5  kilometers  ^ 

7.5 


400075)3001900.0 
2800525 


2013750 
2000305 


38.   A  minute  is  60  seconds. 
How  many  miles  and  how  many 
kilometers      can     light     travel 
through  air  in  a  minute? 
300190  km. 


18,011,400  km. 

186,500  mi. 
60 


11,190,000  mi. 

39.    An   hour  is    60   minutes. 
How  many  miles  and  how  many 


kilometers  can  light  travel  in  an 
hour? 

18,011,400  km. 
60 


1,080,684,000  km. 

11,190,000  mi. 
60 


671,400  000  mi. 

40.  The  distance  round  the 
earth,  given  in  Ex.  37,  is  meas- 
ured on  a  north  and  south  line. 
Around  the  equator  the  distance 
is  40,075.45  kilometers.  How 
many  times  could  light  move 
round  the  equator  in  one  min- 
ute ?      . 

7.49 

400754  5):5001 9000.00 
28052815 

19661850 
16030180 

36316700 
36067905 

7.49 
60 


449. 


41.    Find  the  reciprocal  of  tlie 
between    31.24    and 


difference 
31.23768. 


31.24 

31.23768 

0.00232 


50 


ARITHMETIC. 


431.034 
232)100000.000 
928 


720- 
696 

240 
232 

800 
690 

1040 
928 

42.  The  Hanoverian  mile  is 
25,400  Hanoverian  feet  long, 
each  foot  being  0.9542  of  an  Eng- 
lish foot.  Find  to  four  places  of 
decimals  the  fraction  that  an 
English  mile  of  5280  English 
feet  is  of  a  Hanoverian  mile. 
0.9542 
25400 


3816800 
47710 
19084 

24236.6800 

0.2178 
2423668)528000.0000 
4847336 


4326640 
2423668 

19025)720 
16965(>76 

21610440 
19389*44 


43.  Express  in  inches  the 
length  of  a  meter,  given  that  a 
meter  is  one  ten- millionth  of  a 
quarter  of  the  earth's  circumfer- 
ence, that  the  circumference  is 
3.14159  times  the  diameter,  that 
the  diameter  is  7911.7  miles,  and 
that  a  mile  is  5280  x  12  inches. 

5280 
12 

10560 
5280 

63360 
7911.7 

443520 
63360 
63360 
570240 
443520 

501285312. 
3.14159 

4511567808 
2506426560 
501285312 
2005141248 
501285312 
1503855936 
4)1574832923.32608 
393708230.83152 
0.0000001 

39.370823083152 
-  39.3708  in.  Aiu. 

44.  How  must  a  number  be 
altered  to  double  its  reciprocal  ? 

Divided  by  2. 


teachers'  edition.  51 

.    45.   What  effect  is  produced  on  the  sum  of  two  numbers,  if  each 
number  is  increased  by  the  same  number  ?   What  effect  on  the  differ- 


ence 


It  is  increased  by  two  times  the  number ;  not  any. 

46.  What  effect  is  produced  on  the  product  of  two  numbers,  if 
both  numbers  are  multiplied  by  the  same  number  ?  What  effect  on 
the  quotient*^ 

It  IS  multiplied  by  the  square  of  the  number  ;  not  any. 

47.  What  effect  is  produced  on  the  remainder,  if  both  divisor  and 
dividend  are  multiplied  by  the  same  number?  If  both  are  divided 
by  the  same  number  ? 

It  is  multiplied  by  the  number  ;  it  is  divided  by  the  number. 

48.  In  going  from  one  planet  to  another  light  probably  moves 
faster  than  in  air.  Suppose  it  moves  at  309,800  kilometers  a  second, 
how  long  would  it  take  light  to  perform  each  of  the  following  journeys : 

Moon  to  Earth 375,500  kilometers. 

Sun  to  Karth 147.250,000 

Sun  to  Mercury 56,900,000 

Sun  to  Venus 106,400,000 

Sun  to  Mars 224.100,000 

Sun  to  the  Asteroids 400,000,000 

Sun  to  Jupiter 765,400,000 

Sun  to  Saturn 1,403,000,000 

Sun  to  Uranus 2,817,000,000 

Sun  to  Neptune 4,421,000,000 

Sun  to  the  nearest  star  .     .      24,000,000,000,000 

1.21  475.3  183.7 

3098)3755.00  3098)1472500.0  3098)  569000.0 

3098  12392  3098 

6570  23330  25920 

6196  21686  24764 

3740  16440  11560 

3098  15490  9294 

9500  22660 

9294  21686 


52 

ARITHMFTIC. 

343.4 

723.4 

1291.1 

3098)1064000.0 

3090) 

2241000.0 

3098)4000000jO 

9294 

21686 

3098 

13460 

7240 

9020 

12392 

6196 

10440 

6196 

10680 

28240 

9294 

9294 

27802 

13860 

12460 

3580 

12392 

12392 

3098 

4820 
3098 

2470.6 


3098)7654000.0 
6196 

14580 
12392 


4528.7 


3098)14030000.0 
12392 

16380 
15490 


9092.9 


3098)28170000.0 

27882 


28800 
27882 


21880 
21686 

8900 
6196 

9180 
6196 

19400 

18588 

27040 

24784 

29840 

27882 

22560 
^1686 

14270.5 

77469335 

3098)44210000.0 
3098 

3098) 2 
2 

40000000000 
1686 

13230 
12392 

23140 
21686 

8380 
6196 

14540 
12392 

10380 
9294 

21840 
21686 

21480 
18588 

10860 
Q294 

15400 
15490 

28920 

27882 

15660 
15490 

teachers'  edition.  53 


49.    A  kilometer  is  about  0.6214  of  a  mile.  How  many  miles  is 
each  of  the  planets  from  the  sun  ? 

14725  5690  10640 

6214  6214  6214 


58900 

22760 

42560 

14725 

5690 

10640 

29450 

11380 

21280 

88850 

34140 

63840 

b,  91, 50],  150 

Mercury,  35,357,660 

Venus,  66,116,960 

22410 

76540 

6214 

6214 

89640 

306160 

22410 

76540 

44820 

6214 

153080 

134460 

40000 
Asteroids,  248,560,000 

459240 

,  139,255,740 

Jupiter,  475,619,560 

140300 

281700 

442100 

6214 

6214 

6214 

561200 

1126800 

176S400 

140300 

281700 

442100 

280600 

563400 

884200 

841800 

1690200 

2652600 

Saturn,  871,824,200    Uranus,  1,750,483,800    Neptune,  2,747,209,400 


Exercises  on  Page  73. 

1.  ('onvc'i-t  5427""  into  kilometers;    into  millimotei's  ;    into  centi- 
iiieLei  s. 

5427"'  =  5.427''"'  =  5427000°'"'  =  542700°°'. 

2.  6853""'"  contain   how   many  meters  ?    how  many  centimeters  ? 
what  part  of  a  kilometer  ? 

6853«"'»  =  6.853««  -  685. 3«"'  =  0.006853'"»\ 


54 


ARITHMETIC. 


3.  Write  49.7"'  as  centimeters  ;  as  millimeters  ;  as  part  of  a  kilo- 
meter. 

49.7'°  =  4970""  =  49700™"  =  0.0497"™. 

4.  How  many  centimeters  in  12.4*""?  how  many  millimeters? 
12.4km  _  1,240.000"°  =  12,400,000™"'. 

5.  Change  1280  meters  into  kilometers ;  into  centimeters. 
1 230™  =  1 23''™  =  1 23,000™. 

6.  Write  1230«™  as  meters  ;  as  millimeters. 
1230c™  =  12.3™  =  12,300™™. 


7.  0.435"'  +  852c™  ^  4263™™  + 
0.1595"™. 

0.435™ 
8.52 
4.26 
159.5 


172.718^ 


8.   0.927''™  -  6495c™  .  4.37cm  _ 
42.87™™. 

927.™  0.0437™ 

64.95  0.04287 


862.05™ 


0.00083™ 


9.   8x0.0457"™;  3.04x60.93c™; 
5.43  X  67.2™™. 

0.0672™ 
0.6093™  5.43 

3.04  15T6 

45.7™  24372  2688 

8  18279  3360 


365.6" 


1.852272™    0.364 H96™ 


10.  38,OI9n>"'-- 0.097:  0.41"™ 
+  25.625. 


16" 

25625)410000 
25625 


1437r>0 
143750 


391.948« 
97)38019.000 
291 
891 
873 

189 
97 

920 

873 

470 
388 

820 
776 


11.   At  $1.87  the  met^r  what 
is  tlie  cost  of  6.20™  of  clotli ".' 
11.87 
6.2 
"374 
1122 


111.594 
111.59.  Ans. 


TEACHERS     EDITION. 


65 


12.   At  10.75  the  meter  what  is 
the  cost  of  GO""  of  cloth  ? 
$0.75 


$45.00 


13.  From  a  piece  of  cloth  con- 
taining 47.60™  a  tailor  cuts  off 
three  pieces :  the  first  of  3.80™, 
the  second  of  1.30™,  and  the  third 
of  45«™.  How  much  of  the  cloth 
is  left  ? 


3.8™ 

1.3 

47.6™ 

0.45 

5.55 

5.55™ 

42.05™ 

14.   What 

is  the  value  of  60«™ 

of  cloth,  worth  $5.20 

a  meter  ? 

$5.20 

0.6 

$3.12     . 

15.  If  $6.00  are  paid  for  a 
railroad  ticket  to  travel  440"^™, 
what  is  the  fare  per  kilometer  ? 

0.0138  =  $0,014 
440)  6.000 
440 

1600 
1320 

3800 
3520 

280 

16.  If  a  train  run  288^^™  in  9 
hours,  how  many  meters  does  it 
run  in  a  minute? 


60 

533.33' 

9 
540 

54)  28800.00' 
270 

180 

162 

180 

162 

180 

162 

180 

162 

17.  If  a  man  walk  at  the  rate 
of  6^™  an  hour,  what  part  of  an 
hour  will  it  take  to  walk  420 
meters  ? 


6km 


=  6000™  0.02 

6000)420.00 

42000 


18.  A  railroad  carried  412 
passengers  18  kilometers,  and 
received  $88,992;  at  the  same 
rate,  what  will  it  receive  for 
carrying  350  passengers  35  kilo- 
meters ? 


412 

0.012 

18 

7416)88.992 

3296 

7416 

412 

14832 

7416 

14832 

350- 

12250 

35 

$0,012 

1750 

24500 

1050 

12250 

12250 


$147,000 


56 


ARITHMETIC. 


Exercises  on  Pages  75  and  76. 


1.  Convert  1,854,2761°'  into 
hektars  ;  into  square  kilometers. 

l,854,276i»  =  185.4276h»; 
=  1.8542761'"". 

2.  How  many  hektars  in 
2.7856  square  kilometer,-? 

2.78561'""  =  278.56''a. 

3.  Write  1.7431<i"»  as  square 
centimeters ;  as  square  milli- 
meters. 

1.7431qm  _  17431qcm  . 

=  1,743,1001""". 

4.  How  many  square  kilo- 
meters in  17,467.5  hektarn  ? 

17,467.5'>»  =  174.6751''™. 


5.    How  many  square  meters 
in  1.36141'"°  ? 

1.36141'^'"  =  1,361,4001"'. 


6.    How  many  square   meters 
in  2.25  hektars  ? 

2.25h»  =  22,5001"'. 


7.  How  many  square  centi- 
meters in  0.0137  of  a  square 
meter  ? 

0.01371"  =  1371"'". 

8.  Write  3.571i'='"  as  souare 
millimeters. 

3  571qom  ^  357,lqmin 


Exercises  on  Page  77. 


1.    How    many    cubic    centi- 
meters in  2.25<'»>'»? 

2.25°'>™  =  2,250,000"'". 


2.   How  many  cubic  meters  in 
2,162,875''«»? 

2,162,875«'"«  =  2.162875*'"". 


Exercises  on  Page  78. 


1.  How  many  liters  in  1.7«'""? 
in  157,854««'»  ? 

1.7"^'"  =  1700'; 
157,854«<='»  =  157.854>. 

2.  How    many    cubic     centi- 
meters in  9.5»?   in  0.015»? 

9.5>  =  9500««'» ; 
0.015^=  15«5». 


3.  Change  1.25'''  to  cubic  cen- 
timeters ;  to  the  fraction  of  a  cubic 
meter. 

1.25"=125,000««»; 
—  0.125«'»"'. 

4.  Convert  431.88»  into  hekto- 
liters  ;  into  the  fraction  of  a  cubic 
meter. 


TEACHERS     EDITION. 


57 


431.881  =  4.3188'^i; 
=  0.43188<=i"". 

5.   Write  0.375"^™  as  liters  ; 
cubic  centimeters. 

0.375«bm  _  3751 . 

=  375,000'''"». 


6.  Write  734,159.651'=«'«  as  lit- 
ers ;  as  hektoliters ;  as  cubic 
meters. 

734,159.651<'<"»  =  734.1596511 ; 
=  7.34159G51hi; 
=  0.734159651 


7.  How  many  meters  in 
8,573,412.867°°™? 

8,573,412.867°<=°»  = 

8.573412867°"". 

8.  Change  the  expression 
0.734578912°''«»  into  cubic  centi- 
meters ;    into  liters. 

0. 73457891 2°i"«  =  734,578.912°°'^ ; 
=  734.5789121. 

9.  Change  1731.5  liters  into 
cubic  meters  ;  into  cubic  centi- 
meters. 

1731.51  =1.7315°bn>; 

=  1,731,500°°°\ 


Exercises  on  Page  79. 


1.   How  many  kilos  in  1.73*? 
in  0.341  of  a  ton  ? 


1.73*=  17301^8 ; 

0.341*=  341''g. 

2. 

How    many"  kilos 

will 

a 

hektoliter  of  water  wei^ 

;h? 

100^8. 

3.  Convert  13,756'»k  into 
grams ;  into  the  fraction  of  a 
kilo. 

13,756'°8=  13.7566; 

=  0.0137561^8. 


4.  What    is    the    weight     in 
grams  of  346.1°°™  of  water? 

346.18. 

5.  Give   the   weight   in   kilo- 
grams of  0.37615°i>™  of  water. 

376.15kg. 

6.  Change  0.67781^8  into  milli- 
grams. 

677,800™K.. 

7.  How  many  milligrams  in 
the  third  part  of  17.4  grams? 

5,800™8. 


58 


ARITHMETIC. 


Exercise  IV. 


1.   Add  17.3™,  87.41' 

SSO""™,  and  1.79™. 

17.3» 

87.41 

2.71 

0.38 

1.79 


271* 


109.59^ 


2.  What  is  the  sum  of  $15.87, 
$39.46,  $47.52,  $75.38,  $75.89? 
$15.87 
39.46 
47.52 
75.38 
75.89 


$254.12 

3.   Add   187°",  49.3™,   317" 
and  6.138™. 

1.87™ 
49.3 
0.317 
6.138 


57.625™ 


4.  The  door-sill  being  3«™ 
high  ;  the  door,  2.34™  ;  the  finish 
over  it,  13.7"™  ;  and  the  distance 
from  finish  to  coiling,  930™ .  how 
far  from  floor  to  ceiling  ? 

0.03« 

2.34 

0.137 

0.93 

3.437« 


5.  The  distance  to  the  post- 
office  is  3.31^^™ ;  thence  to  the 
mill,  1.71P™ ;  thence  to  the  store, 
3.718^™;  thence  home,  2.543''™. 
How  long  is  the  circuit? 

3.31"™ 

1.711 

3.718 

2.543 


11.282''" 


6.  From  Portland,  Me.,  to 
Boston  is  about  132"™ ;  Boston 
to  Albany,  320"™;  Albany  to 
Bufi'alo,  480"™  ;  Buffalo  to  Chi- 
cago, 800"™  ;  Chicago  to  Omaha, 
800"™;  Omaha  to  Cheyenne, 
780"™ ;  how  far  from  Cheyenne 
to  Portland  ?  to  Albany  ?  from 
Boston  to  Chicago  ?  from  Boston 
to  Cheyenne  ? 


(1) 

(2) 

132"™ 

132km 

320 

320 

480 

452"™ 

800 

800 

3312"™ 

780 

452 

3312"™ 

2860"™ 

(3) 

(4) 

320"« 

3312"™ 

480 

132 

800 

3180"" 

1600"' 


TEACHERS     EDITION. 


59 


7.   If  I  travel  789.7"'^  a  day,  how  far  shall  I  go  in  7  days  ?  in  8.5? 
in  19.6  ?  in  27.8  ?  in  365  days  ? 

789. 7'^'^ 

7 

789.7>^-            789.^ 
8.5                  19.( 

39485               4738^ 
63176              71073 

^km             789.7'^°» 

5                   27.8 

}                63176 
55279 
15794 

789. 7'^'" 
365 

5527.9'^'^ 

39485 

47382 

6712.45'^'"        "^^^^ 
15478.1 

23691 

2^-^        21,953.66'^- 

288,240.5'''»' 

8.   How  m 
cost,  at  $1.^ 

uch  will  3"'  of  cloth 
J7  a  meter?     How 
at  $2.63  a  meter? 

$4,875 
153 

much  5.38'", 

14625 

$1.37 
3 

$2.63 
5.38 

24375 

4875 

$4.11 

2104 

789 

$745,875 
=  $745.88. 

Ans. 

1315 


$14.1494 
=  $14.15.  Ans. 

9.  How  much  will  13.4i'g  of 
opium  be  worth,  at  $  8.48  a  kilo  ? 
28.79i^«,  at$7.96akilo? 


$8.48 
13.4 

3392 
2544 
848 


$113.63 


28.79 
$7.96 

17274 
25911 
20153 

$229.1684 
=  $229.17.  Ans. 


10.  A  man  bought  153  barrels 
of  flour,  at  $4,875  a  bbl.  What 
did  the  whole  cost  him  ? 


11.  He  gave  for  it  6  shares  of 
stock,  at  $113.50  a  share,  and 
the  rest  in  cash.  How  much 
money  did  he  pay? 

$113.50 
6 


$681.00 

$745.88 
681. 

$  64.88 


12.  He  paid  $13.75  for  stor- 
age ;  also,  75  cents  a  barrel  for 
freight.  How  much  do  these 
expenses  amount  to  ? 


60 


ARITHMETIC. 


153 

$075 

765 

1071 

$114.75 

13.75 

$  128.5a 

13.   It  was  sold,  49  bbls.  at 
$6.50  a  bbl.;  the  rest  at  |6.25. 
"What  were  the  gross  receipts? 
$6.50  $6.25 

49  104 


5850 
2600 
$318.50 
153 
_49 
104 


2500 

625 
$650.00 

318.50 
$968.50 


14.  He  paid  for  commissions, 
etc.,  $17.50;  and  counts  his  los-s 
of  interest  at  $29.30.  What  then 
is  his  net  profit  ? 

$745.88 

128.50 

17.50 

29.30 


$921.18 

$968.50 
921.18 

$  47.32 

15.  Find  the  circumference  of 
a  circle  having  a  diameter  of 
1"». 

3.1416 


3.1416- 


16.   Find  the  circumferences  of  circles  of  which 
respectively  83'";  3.71'°;  32.8°';   10.4<"° ;   11.8<='° 
Give  each  to  the  nearest  tenth  of  a  millimeter. 
3.1416  3.1416 

83000'°'°  3710°'™ 


the  diameters  are 
;  167.1°'°';  39.3°'°'. 

3.1416 
32800°'°' 


94248000 
251328 
260752.8°'°' 


314160 
219912 
94248 


25132800 
62832 
94248 


11655.3°' 

m 

103044.4800°'°' 

= 

103,044.5°'°'.  Ans. 

3.1416 

3.1416 

3.1416 

3.1416 

104mm 

118°'°' 

167.1°'°' 
31416 

39.3°"' 

125664 

251328 

94248 

31416 

31416 

219912 

282744 

326.7264™°' 

31416 

188496 

94248 

326.7°"°.  Ans. 

370.7088°'°' 
=  370.7°'°'.  Ans. 

31416 

123.46488°'°' 

524.96136°'°'  =  123.5-°'.  Ans. 

= 

=  525°«'.  Ans 

TEACHERS     EDITION. 


Gl 


17.   What  is  the  length  of  the 

(2) 

earth's    orbit,    to    the    nearest 

=  4.115"^  Ans. 

meter,   if   the    diameter  of  the 

orbit  is  294,481, 21 7'^'^? 

(3) 

294481 21 7'^'^ 

Ills'" 

3.1416 

17 

1766887302 

28805 

294481217 

4115 

1177924868 

69.955'n.  j{ns. 

294481217 

883443651 

20. 

How  often  must  that  wheel 

925,142,191.3272'^'» 

turn  in  going  69.429'"?  73.5 13"^  / 

-925,142,191,327'".  Ans. 

17.2? 

on? 

18.    How  far  round  this  world. 

(1) 
17  nearly 

if  its  diameter  is  12,734'™? 

12734''°^ 

4115)69429 

3.1416 

4115 
28279 

12734 

50936 

12734 

38202 

Ans. 

40,005.1344'^"'. 

19. 

If    a    carri  age- 

wheel 

is 

1.31'" 

in   diameter,  what  is 

its 

circumference  ?      How 

far   will 

it  go, 

if  it  roll  without 

slipping. 

in  turning  once?  17  times? 

(1) 

3.1416 

1.31- 

31416 

94248 

31416 

4.115496™ 
4.115™.  Ans. 


(2) 


18  nearly 


4115)73513 
4115 


32363 


(3) 
4197  nearly 


4115)17270000 
16460 


8100 
4115 


39850 
37035 
28150 


21.   Find     the    reciprocal    of 
I   3.14159  to  the  5th  place. 


62 


ARITHMETIC. 


0.31831 

24.   How  thick  through  is  a 

314159)100000.00000 

tree  which  has  a  girth  of  2.97»'? 

942477 

0.31831 

575230 

2.97» 

314159 

222817 

2610710 

286479 

2513272 

63662 

974380 

0.9453807» 

942477 

=  0.915'".  Am. 

319030 

314159 

25.   What  is  the  diameter  of  a 

circular  field  two  kilometers  iu 

circumference  ? 

22.   What  is  the  diameter 

of 

the  circle  whose  circumference 

is 

0.31831 

314.159'""? 

2000 

lOQcm 

636.62" 

314159)31415900 

314159 

26.   What  is  the  diameter  of  a 

rope  of  which  the  circumference 

23.   What  is  the  diameter 

of 

is  20"'"? 

the  wheel  which  revolves   1£ 

.5 

0.31831 

times  in  going  107.25">? 

20cm 

5.5"» 

6.3662«»» 

195)1072.5'^ 

975 

27.   In  a  park  is  a  fountain 

975 

whose  basin  is  75""  in  circuni  er- 

975 

ence.     What  is  the  diameter  of 
the  basin  ? 

0.31831 

0.31831 

5.5« 

75'" 

159155 

159155 

159155 

222817 

1.750705"' 

23.87325'" 

=  1.75'".  A718. 

-  23.87"'.  Am. 

TEACHERS     EDITION. 


63 


Exercise  V. 

1.   Find  the  area  of  a  rectangle  17"™  by  19«". 
19cm 


133 
19 


323i« 


2.  In  a  rectangular  township  16^  by  7^™,  how  many  hektars  ? 
If  there  are  in  it  47.3^™  of  highway,  averaging  11.7™  wide,  how 
much  land  is  left  for  other  uses  ? 

47300™ 
11.7 


331100 
473 
473 

5534101™ 
55.34  P* 


11144.659»'» 


3.  In  a  rectangular  field,  751.3™  long  and  189.3™  wide,  is  a  straw- 
berry bed  31.4™  by  17.8™.  How  many  hektars  in  the  field?  How 
many,  exclusive  of  the  strawberry  bed  ? 

751.3™  31.4™ 

189.3™  17.8™ 


22539 

2512 

67617 

2198 

60104 

314 

7513 

558.921™ 

142221.091™ 

-  0.056'>» 

:  14.222'»^  Ans. 

14.2221'* 

0.056»"' 

Ans. 

14.1661'*. 

64 


ARITHMETIC. 


4.   If  my  garden  contain  941.65«",  and  my  neighbor's  748.37*i"', 
what  is  the  area  of  both  in  hektars  ? 

941.65<i°»  =  0.094 165»'» 
748.371'"  =  0.074837*" 


0.169002''* 
=  0.169»'».  Am. 

5.   If  a  painter  can  cover  8.786i°'  in  an  hour,  how  much  can  he 
cover  in  1.78  hours?  in  3.86  hours  ?  in  4.57  hours  ? 


8.7861'" 

8.7861"" 

8.7861'" 

1.78 

3.86 

4.57 

70288 

52716 

61502 

61502 

70288 

43930 

8786 

26358 

35144 

15.6391'" 

33.913961'" 

40.152021"' 

=  33.9141'".  Ans. 

=  40.1521"'.  Ans. 

6.  How  many  hektars  in  each  of  three  rectangular  fields :  one 
measuring  315.71'"  by  78.91"';  a  second,  293.6"'  by  84.84'";  the  third, 
346.8"*  by  71.82"'.     How  many  in  the  three  ? 

315.71"'  293.6'"  346.8"' 

78.91"*  84.84"*  71.82" 


31571 

11744 

6935 

284139 

23488 

27744 

2.4913»'* 

252568 

11744 

3468 

2.4909»'» 

220997 

23488 
24909.0241"' 

24276 
21907.1761"' 

2.4907*'* 

24912.67611"' 

7.4729''».  Ans. 

2.4913"*.  Ans. 

=  2.4909b^  Ans. 

=  2.4907»'».  Ans. 

7.   Give  the  price  of  each  field,  and  of  the  whole,  at  $67.50  a  hek- 
tar  ;  at  $384  a  hektar  ;  and  at  $2,375  a  square  meter. 


2.4913 
$67.50 

1245850 
174391 
149478 


2.4909 
$67.50 

1245450 
174363 
149454 


2.4907 
$67.50 


1245350 

$168.16 

174349 

168.14 

149442 

168.12 

2.4913 
$384 

99G52 
199304 
74739 


$108.162950      $168.135750      $168.122250  $504.42   $956.6592 
=  $168.16. ilns.  =$168.14.  ilns.  =  $168.12.  Ans.  =$956.66  Ans. 


TEACHERS     EDITION. 


65 


8.  What 

is  the  area  of  a  circle 

784n> 

614656'i°» 

27««'  in  diameter? 

of  one  which 

784m 

0.7854 

is  1°^  in  diameter  ? 

3136 

2458624 

(1) 

6272 

3073280 

27cm 

0.7854 

729qcm 

70686 

5488 

4917248 

27cm 

614656*1'° 

4302592 

189 

482750.8224^" 

54 

15708 
54978 

="48.275h^  Arts. 

729qcm 

572.5566<i''«^ 

10.   Give  the 
31"'"  in  diameter 

area  of  a  circle 

(2) 

31cm 

0.7854 

im  X  Im  X  0.7854 

=  0.7854^'- 

31cm 

961qcm 

31 

7854 

9.  What 

is  the  area  in  hek- 
circular    field    784"" 

93 

47124 

tars    of   a 

961qcm 

70686 

across  ? 

754.7694i«°» 

11.   Find  the  length  of  a  rectangle  17*'™  wide,  and  containing 
306<icm.     \Vhat  length  of  carpet  75«'°  wide  is  required  to  make  27**°^  ? 
18«°»  ^Q^ 

17)306  75)  2700 

17  225 


136 
136 


450 
450 


12.  A  room  is  16"°  long,  8"  wide,  and  8™  high  ;  another  room  is 
7™  long,  7''*  wide,  and  3™  high.  How  many  square  meters  of  paint- 
ing on  the  walls  of  both  rooms,  if  no  allowance  is  made  for  doors 
and  windows  ?  How  many  more  square  meters  of  painting  on  the 
walls  of  the  larger  room  than  on  those  of  the  smaller  ? 
7m  14m  iQm  24™  3841™  384^™ 

7  2  8  2  84  84 


14r 


24^ 


48 


468<i™.  Ans.      300<i™.  Ans. 


84qm 


3841™ 


66 


ARITHMETIC. 


13.  How  many  square  centi- 
meters of  surface  on  a  ball  7*"°  in 
diameter  ? 


70m 

49qcni 


3_1416qcm 

49 


282744 
125664 
153.93841°°' 


14.  How  many  square  centi- 
meters of  surface  on  a  ball  18°™ 
in  diameter? 


18«a 

144 
18 

324qcm 


3.1416 
324qcm 


125664 
62832 
94248 
1017.8784i«'° 


15.  How  many  square  meters 
of  surface  on  a  hemispherical 
dome  11.27™  in  diameter? 


11.27™ 
11.27°' 
7889 
2254 
1127 

1127 

1 27.01 29<»° 


127.0129^™ 

3.1416 
7620774 
1270129 
5080516 
1270129 
3810387 
2)399.023726641™ 
199.511863321™ 
=  199.51191™.  Ans. 


16.  What  ifl  the  interior  sur- 
face of  a  hemispherical  basin 
12<=™  in  diameter? 

12°™  3.1416 

1441°™  125664 

125664 
31416 
2)452.39041°™ 
226.19521°™ 

17.  What  is  the  interior  sur- 
face of  a  hemispherical  vase  70°™ 
in  diameter  ? 

70°™  3.1416 

70°™  49001°™ 

49001°™  27274400 

125664 
2)15293.84001^ 
7641.921°™ 

18.  Find,  by  this  rule,  the. 
area  of  example  9. 


2)784 
392 
392 
784 
3528 
1176 
1536641™ 


1536641™ 
3.1416 
921984 
153664 
614656 
153664 
460992 

482750.82241™ 
48.275"*.  Ans. 


19.  How  many  square  centimeters  are  inclosed  in  a  circle  struck 
with  a  radius  of  7°™? 

7°™  3.1416 

tjatn  49qom 

491^  272744 

125664 


152.9384i°«'.  Ans. 


TEACHERS     EDITION. 


67 


20.  In  a  sheet  of  zinc  1.76™  long  and  89<'™  wide  are  two  circular 
openings,  one  of  which  has  a  radius  10.5"'",  the  other  a  radius  9.2<=™. 
What  is  the  area  of  the  zinc  left  ? 

3.1416  9.2«'« 


10.5« 


10.5«™ 

110.251°™ 

9.2cm 

525 

157080 

184 

105 

62832 
31416 
31416 

828 

110.25i«'» 

3464qcm 

346.36140010™ 

=  0.034641™ 

3.1416 

0.026591™ 

1.76™ 

84.64  vm 

0.03464i™ 

0.89™ 

125664 

0.061231™ 

1584 

188496 

1408 

125664 
251328 

1.56641™ 
0.061231™ 

265.905024i«'° 
0.02659^™ 

1.505171™ 

21.  Whatis  the  area  of  a  circle 
of  which  the  radius  is  24™? 


24™ 
24™ 

96 

48 

5761" 


3.1416 

5761™ 

188496 
219912 
157080 

1809.56161™ 


22.  A  piece  of  land  in  the  form 
of  a  circle  has  a  radius  of  40™ ; 
in  the  middle  of  it  is  a  pond  form- 
ing a  circle  of  15™  radius.  What 
is  the  total  surface  ?  the  surface 
of  the  pond  ?  the  surface  of  the 
land  to  cultivate  ? 


3.1416 
2251™ 

157080 
62832 
62832 


40™ 
40™ 

16001' 


706.861™,  surface  of  pond. 

3.1416 
16001™ 


18849600 
31416 


5026.561™,  total  surface. 
706.861™ 


4219.71™,  surface  of  land. 


68 


ARITHMETIC. 


24.   How  many  meters  of  car- 
pet 56°™  wide  will  be  required  for 
a  room  8.32™  long  and  6.6™  wide, 
strips  running  lengthwise  ? 
11.7  =  12  strips. 
56)660.0 
56 


100 
56 

440 
392 

48 


8.32™ 
12 

1664 
832 

99.84^ 


25.  How  many  meters  of  car- 
pet 70*=™  wide  will  be  required  for 
a  room  7™  long  and  5.4™  wide, 
strips  running  across  the  room  ? 

10  strips. 
70)700  5.4™ 

70  10 

0  54°' 

26.  How  many  meters  of  car- 
pet 80°™  wide  will  be  required  for 
a  room  6™  long  and  5.47™  wide, 
strips  running  across  the  room  ? 

7.5  strips  =  8  strips. 

sojeoo 

560  5.47"' 

400         ?_ 

400         43.76™ 


27.   How  many  meters  of  car- 
pet 90°™  wide  will  be  required  for 


a  room  5™  long  and  4.5™  wide, 

strips  running  lengthwise?  How 

much  will  it  cost,  at 

$1,875  a 

meter  ? 

5  strips. 

5» 

90)450 

5 

450 

25» 

$1,875 

25 

9375 

3750 

$46,875 

=  $46.88 

28.  How  many  meters  of  car- 
pet 75°™  wide  will  be  required 
for  a  room  5.25™  long  and  4.75™ 
wide,  strips  running  across  the 
room  ?  How  much  will  it  cost, 
at  $2,125  a  meter? 


7 

strips. 

4.75™ 

75)525 

7 

525 

33.25™ 
$2,125 

16625 

6650 

3325 

6650 

$70.65625 

B 

=  $70.66 

29.  How  many  meters  of  car- 
pet 75°™  wide  will  be  required  for 
a  room  5.6™  square?  How  wide 
a  strip  will  have  to  be  turned 


TEACHERS     EDITION. 


69 


under  ?    How  much  will  the  car- 
pet cost,  at  $1.25  a  meter? 

7.4  =  8  strips.  5.6^ 

75)560.0  ^ 

525  44. 8"! 

"^  |1;25 

300  2240 
896 

75cm  448    ■- 

^  156.00 

40°™  to  turn  under. 


31.  How  many  rolls  of  paper 
45cm  ^jffi^Q  and  8™  long,  allowing 
11. 19*1™  for  doors  and  windows, 
will  be  required  to  paper  a  room 
whose  length  is  6.12'°,  breadth 
5.05^,  and  height  3.5™  ? 


78.19^™ 
11.19^™ 


eyqm 


0.45 


3.601™ 


18.6  =  19  rolls. 


36)670.0 
36 

310 

288 

220 
216 


32.  Find  the  cost  of  papering 
a  room  8™  long,  5.5™  wide,  and 
4.5™  high,  with  paper  50°™  wide 
and  7.5™  in  a  roll,  at  $1.25  a 


roll,  put  on  ?  There  is  a  base- 
board 25°™  wide  running  round 
the  room,  and  an  allowance  of 
11*1™  is  made  for  doors  and  win- 
dows. 


8™ 

27™ 

5.5 

0.25™ 

13.5™ 

135 

2 

54 

27™ 

6.751™ 

4.5™ 

11 

135 

17.751™ 

108 

121.51™ 

7.5™ 

17.75 

.5™ 

103.751™ 

3.75i« 

27.6  = 

=  28  rolls. 

375)10375.0 

$1.25 

750 

28 

2875 

1000 

2625 

250 

2500 

$35.00 

2250 

33.  Find  the  cost  of  plaster- 
ing this  room,  at  $0.50  a  square 
meter. 

5.5™  103.751™,  walls. 

gm  441™ 


441™,  ceiling.    147.751™ 
$0.50 

$73.8750 
=  $73.88 


70 


ARITHMETIC. 


34.  Find  the  cost  of  papering  a  room  5.5™  long,  4.8™  wide,  and 
3.2™  high,  with  paper  45«'n  wide,  7.5"  in  a  roll,  at  $0,875  a  roll,  put 
on,  allowing  12i™  for  base-board,  doors,  etc. 

15.9  =  16  rolls. 


5.5™ 

75™ 

3375)539200 

$0,875 

4.8™ 

0.45™ 

3375 

16 

10.3™ 

375 

20170 

5250 

2 

300 

16875 
32950 

875 

20.6™ 

3.375<i™ 

$14.00 

3.2™ 

30375 

412 

618 

I 

65.92<i"' 

12 

allowance. 

53.92<t™ 


35.   Find  the  cost  of  plastering  this  room,  at  $0.45  a  square  meter. 


5.5™                  26.40<i™,  ceiling. 
4.8™                  53.92,  walls. 

440                   80.32 
220 

80.32 
$0.45 

40160 
32128 

26.4m™,  ceiling. 

$36.14 

36.  Find  the  cost  of  papering  a  room  6™  square  and  3.5™  high, 
with  paper  45"™  wide  and  7.5™  in  a  roll,  at  $0.75  a  roll,  put  on  ;  and 
of  putting  on  a  border,  at  5  cents  per  running  meter. 


6™ 
6™ 

12™ 
2 

24™ 
3.5™ 

120 
72 

34qin 


7.5™ 

0.45™ 

375 
300 

3.375«»™ 


24.8  =  25  rolls. 
3375)84000.0 
6750 


16500 
13500 

30000 
27000 


24 

$0.05 

$1.20 


25 
$0.75 

125 

175 

$18.75 
1.20 

$19.95 


TEACHERS     EDITION. 


71 


37.   Find  the  cost  of  plastering  this  room,  at  |0.36  a  square  meter. 

e-*  $0.36 

Se^"*,  ceiling.  720 

84^1™,  walls.  3()0 


120^™ 


13.120 


38.  Find  the  cost  of  papering  a  room  13™  long,  12"^  wide,  and  7'° 
high,  with  paper  45*"*  wide  and  7.5°^  in  a  roll,  at  |1.50  a  roll,  put 
on  ;  and  of  putting  on  a  border,  at  $0.30  a  running  meter,  allowing 
15qm  foj.  base-boards,  doors,  etc. 


13'" 
12'^ 

7.5°' 
0.45"' 

S75 
300 

3.375^'" 

$ 
allowance. 

0.30 
50 

69.9  = 

=  70  rolls. 

25°' 
2 

3375)  235000.0 
20230 

50- 

7m 

33700 
30375 

33250 
30375 

$1.50 
70 

115i«', 

$  105.00 
15. 

235^'" 


$15.00 


39.    Find  the  cost  of  plastering 
this  room ,  at  $  0.60  a  square  meter. 
13'° 
12°' 


1561°',  ceiling. 
235^°',  walls. 

3911°' 
0.60 


$234.60 

40.  How  many  meters,  board 
measure,  in  a  board  8°'  long  and 
20«°'  wide  ? 

8°» 
0.2°' 

1.61°' 


41.   How  many  meters,  board 
measure,  in  a  joist  5™  long,  25"°' 
wide,  and  75°'°'  thick  ? 
0.25"' 

5m 


1.251°' 
3 

3.751"' 


42.   How  many  meters,  board 
measure,  in  a  stick  of  timber  U")'" 
long  and  40°"'  square  ? 
16 
15  25)400  16 

0.4  25  6 


150 
150 


96 


72 


ARITHMETIC. 


43.  How  many  meters,  board 
measure,  in  2  joists  5™  long, 
27.5<«^  wide,  and  50"^  thick? 

0.275 
5 


1.375 
2 

2.75 
2 


5.5 

44.  How  many  meters,  board 
measure,  in  10  planks,  each  4"* 
long,  45<"«  wide,  and  10<"°  thick  ? 
and  what  is  the  value  of  these 
planks,  at  $25  a  hundred  meters? 

4 

0.45  25)l00 

4  100 


1.80 
4 

10 

72 


72 
$0.25 

360 
144 

$18. 


45.  How  many  meters,  board 
measure,  in  25  box  boards,  each 
4"'  long,  42°'»  wide,  and  20™" 
tliick?  and  what  is  their  value, 
at  $14  a  hundred  meters  ? 

0.42™ 
1.68<J°» 


$5.88 

46.  Find  the  cost  of  10  joists 
4.5"  long,  10«"  wide,  and  7.5«» 
thick,  at  $11  a  hundred  meters. 

4.5"  3  1.35*"» 

__1"  25)75  10 

0.45qm  75         13.50<»" 

3  $0.11 


LSSi"* 


135 
135 


$1,485 
=  $1.49.  Ans.' 

47.  Find  the  cost  of  36  planks, 
each  4"  long,  27.8""  wide,  and 
75""  thick,  at  $16  a  hundred 
meters. 


0.278" 
4" 

3 

25)75 

75 

120.0961" 
$0.16 

1.112«»" 
3 

720576 
120096 

3.336«»" 
36 

$19.21536 
.$19.22.  A71S. 

20016 
10008 

120.096*1" 

TEACHERS     EDITION. 


73 


48.   Find  the  cost  of  3  sticks 
of  timber,   each  8'"  long,  22.5<'"' 
wide,  and  20""^  thick,  at  $17.50 
a  hundred  meters. 
0.225"*  8 

25J2OO 
200 


8'" 


1.81"' 

8 

14.41"' 
3 

43.21"' 


49.  Find  the  cost  of  a  board 
8.25"'  long,  28°"'  wide  at  one  end 
and  35""'  at  the  other,  and 
31.25"'"'  thick,  at  $0.30  a  meter. 

0.28"*  2.598751"' 

0.35"!  1.25 


43.21"* 

0.30'" 

10 

27.51"* 

$0,175 

0.25"' 

25) 250 
25 

$0.14 

2160 

2)0.55"' 

1100 

3024 

0.275"^ 

0 

275 

432 

10™ 

$3.85 

7.56 

2.751"* 
10 

2.598751"*       =$0.97. 


50.  Find  the  cost  of  a  stick  of 
timber  10"*  long,  25«"'  thick,  30"" 
wide  at  one  end  and  25''"*  wide  at 
the  other,  at  $14  a  hundred  meters. 


27.5q»* 

51.  Find  the  cost  of  the  floor- 
ing for  a  two-story  building  16"* 
by  10.5"* ;  the  flooring  being 
32"'"'  thick,  and  worth  $30  a 
hundred  meters. 


1.28 


10.5'" 


3361"* 


2)0.63"* 
0.315"* 
8.25 

1299375 
519750 
259875 

25) 

32.00 
25 

70 
50 

200 
200 

16"* 

630 
105 

1.28 

2688 
672 

1575 

3.2484375 

$0.30 

10.974531250 

168.01"* 

2 

336 

630 

430.08 

2520 

3361"*    _ 

$0.30 

129.02 


52.  Find  the  cost  of  the  flooring  timbers  for  this  building,  the 
timbers  being  25°"*  by  50"*"*,  and  placed  on  edge  30°"'  apart,  and 
worth  $11.50  a  hundred  meters. 


30cm 

30 

16"* 
0.25"* 

80 
32 

Aqm 

4qm 
2 

gqm 

30 

2401"* 
$0,115 

1200 
240 
240 

$27.60 

5cm 

35)1050 
105 

2 

35qcm 

$55.20 

2401"* 

$27.60 


74 


ARITHMETIC. 


53.  Find  the  cost  of  the  fencing  to  enclose  a  field  150™  long  an  1 
75™  wide  ;  the  posts  are  set  2.5™  apart,  and  cost  $0.25  apiece ;  the 
fence  is  5  boards  high  ;  the  bottom  board  is  SO"",  the  top  board  25*"*, 
and  the  other  three  each  22.5*""  wide,  and  the  boards  cost  1 13.25  a 
hundred  meters. 

30  150»  122.5™ 

25  75"  450"» 

67.5 


122.5 


180 
$0.25 

900 
360 


$45. 


225<»™ 
2 

450qm 


180  posts. 


25)4500 
25 

200 
200 


Exercise  VI. 


61250 
4900 

55i.2r)im 
$0.1325 

275625 
110250 
165375 
55125 


$73.04 
45 

$118.04 


1.    How    many    cubic    centi- 
meters in  a  block  9<""  long  by 


wide,  and  6" 

9cm 

'jom 

g3qcm 


deep? 

63qcin 


378«= 


2.  If  wood  is  cut  into  120°™ 
lengths,  and  a  pile  is  43.7™  long 
and  1.4™  high,  how  many  sters 
of  wood  are  there  in  it  ? 


43.7™ 
1.2™ 

874 
437 

52.44'!™ 


52.44'*™ 
1.4™ 


20976 
5244 

73.416' 


3.  In  a  grain  elevator  is  a  bin 
11.2™  long,  4.34™  wide,  and  2.83™ 
deep.  How  many  hektoliters  of 
grain  will  it  hold  ? 

11.2™      48.608'»™ 
4.34™       2.83™ 


448 
336 

448 

48.6081™ 


145824 
388864 
97216 

137.5()0G4'''>™ 
1375. 6064". 


4.  If  a  liter  of  grain  weigh 
0.81  of  the  weight  of  a  liter  of 
water,  how  much  will  the  grain 
in  that  bin  weigh  ? 


TEACHERS     EDITION. 


75 


1375.6064W 
=  137560.64'^8 
0.81 

13756064 
110048512 

111,424.1184'^g 

5.   A   bin   measuring  16™  by 
9.7'",  and  2.8">  deep,  is  full   of 
oats,  worth  $0.98  a  hektoliter. 
What  is  the  whole  worth  ? 
16°>  4345.6^1 

9.7"»  $0.98 


347648 
391104 


$4258.688 
$4258.69.  Ans. 


112 
144 

155.2<i'" 

2.8°* 

12416 
3104 

434.56''i>°^ 
=  4345.6J>i. 

6.  A  vat  197"°^  long,  87<='»  wide, 
and  63«'»  deep,  holds  how  many- 
liters  ?  What  would  be  the  weight 
of  water  required  to  fill  it  ? 

197cm  17139i''m 

87cm  g3cm 


1379 
1576 
17139qcm 


51417 
102834 

1079757°«"» 

1079.75? 

1079.757'^8. 


7.  Add  1341<=«™,  2311,  and 
2.13*'\  and  give  the  sum  in  terms 
of  each  of  the  three  units. 


134lccm 

2311  =  231000«'™ 

2.13''i  =  213000O'"" 


445341°««' 
=  445.3411 
=  4.45341^1. 


8.  If  a  spring  pours  out  467.8^ 
each  minute,  how  many  hekto- 
liters  will  it  deliver  in  60  min- 
utes ?  in  37  minutes?  in  78 
minutes  ? 

467.81  4.678" 

=  4.678"  37 

60 


280.68" 


32746 
14034 

173.086" 


4.678" 
78 

37424 
32746 

364.884" 

9.   If  67.31  of  oil  in  a  vat  with 

perpendicular  sides  fill  it  to  a 
depth  of  173'°'",  how  deep  will 
13.7  times  that  quantity  fill  it? 
and  how  many  hektoliters  will 
there  be  ? 


173" 
13.7 

1211 
519 
173 


2370.1^ 
2.3701« 


67.31 

=  0.673" 

13.7 

4711 
2019 
673 

9.2201" 


76 

ARITHMETIC 

10. 

Into  a  round   cup    10»" 

tin  cu 

p  95'""»  across  and  11.08<"» 

acroK 

},  with  perpendicular  sides, 

deep? 

pour 

3il  until  it  is  1*^  deep ;  then 

95mm 

=  9.5«»»                 0.7854 

there 

are  78.54«»»  of  oil  in  the 
How   many   cubic    centi- 

9.5<"»                   90.25<J<« 

cup. 

475                     39270 

meters  will  there  be  when   the 

855                   15708 

oil  is 

.38"^  deep  ? 

90.25«»«»           '^^^^^ 

38"™               78.54«^ 

70.88235<»«» 

= 

38cm                            3.8cm 

11.08<"» 

62832 

56705880 

23562 

7088235 

298.452«=«> 

7088235    ' 
785.3764380««"' 

11. 

What  is  the  capacity  of  a 

=  0.785'.  Am. 

12. 

What  are  the  capacities  of  two  cylindrical  vessels,  one  being 

1G.24<"»  across  and  19.95<»°  deep,  th 

e  other 

75  4ram  across  and  87.9"™ 

deep? 

1G.24<''° 

4132.433«°»               75.4'"«» 

1().24<'™            = 

4.132» 

75.4mm 

6496 

3016 

3248 

3770 

9744 

5278 

1624 

5685.16*1"^ 

263.7376'K" 

0.7854 

0.7854 

2274064 

10549504 

2842580 

13186880 

4548128 

21099008 

3979612 

18461632 

4465.124664'«"«' 

207.1395 

-4465.125«»»» 

19.95<« 

87.9°^ 

10356975 

40196126 

18642555 

31255875 

18642555 

35721000 

2071395 

392485.4875«- 

4132.433025«« 

-  0.392'. 

TEACHERS     EDITION. 


77 


13.  How  many  cubic  centi- 
meters in  a  ball  10°™  in  diam- 
eter ?  How  much  less  if  you 
take  the  more  exact  multiplier  ? 


10°™ 
10°™ 

0.5236 

1000°°™ 

1001°™ 
10°™ 

523.6°°™ 

1000°°™ 

'1 

3.1415927 

0.5235988 

1000°°™ 

523.5988°°™ 

523.6°°™ 
523.5988°°™ 

0.0012°°™ 

14.  Into  a  cubical  box  20°™ 
on  a  side,  and  full  of  water,  an 
iron  ball  20°™  in  diameter  is 
gently  lowered  until  it  touches 
bottom.  How  much  water  is  left 
in  the  box?  Answer  in  liters 
and  in  cubic  centimeters. 
20°™  0.5236 

20°™  8000°°™ 


4188.8° 


400i°™ 
20 

8000°°™ 
4188.8°°™ 

3811.2°°™ 
3.81121. 


15.    One  cask  contains  171.41 
of  oil ;  another,  209.3i ;  a  third, 


73.81;  while  a  square  vat,  137°™ 
each  way,  is  filled  to  a  depth  of 


69°™. 


How  much  oil  in  all  the 
?  in  liters  and  in  hekto- 


1 iters. 


137°™             187691°™ 
137°™                   69°™ 

959 
411 

168921 
112614 

137 

1295061°°™ 

18769i<^"'     =1295.0611. 

171.41 

209.31 

73.81 

1295.061J 

1749.5611 
17.49561i'i. 

16.  How  many  liters  of  air  in 
a  room  7.8™  long,  6.23™  wide, 
and  3™  high  ? 

6.23™ 
7.8™ 

4984 
4361 

48.5941™ 
3™ 

145.782°»>™ 
==  1457821. 


17.  If  a  person's  breathing 
spoils  the  air  at  the  rate  of 
0.2175°!'™  a  minute,  how  long 
will  it  take   3   persons   sitting 


78 


ARITHMETIC. 


in  the  room,  closed,  to  spoil  the 

air? 

0.2175 

3 


0.6525 

223.42 
6525)1457820.00 
13050 


15282 
13050 

22320 
19575 


27450 
26100 

13500 


18.   How   long,   at  the  same 
rate,  would  the  air  in  a  hall  22'" 
long,  16"  wide,  and  7°*  high,  last 
an  audience  of  280  persons  ? 
22'»  2175 

16"  280 


132 
22 


352i« 

•7in 

2464«^™ 


174000 
4350 

60.9000 


40.5 


609)  24640.0 
2436 


2800 


19.  How  many  cubic  meters  of  wood  in  a  round  stick  of  equal 
size  throughout,  37*™  in  diameter  and  8.4™  long  ? 


37« 

0.7854 

0.10752126<i» 

37«« 

1369'J°» 

8.4« 

259 

70686 

43008504 

111 

47124 
23562 

86017008 

1369<»«« 

0.903 178584«»>'» 

7854 

=  0.9032«»>'°. 

1075.2 126<i«»» 

-0.10752126«»'». 

Exercise  VII. 

1.  What  is  the  weight,  in  kilograms,  of  a  hektoliter  of  water  ? 
73.8'  of  water  ?  of  a  cubic  meter  of  water  ?  of  a  cubic  centimeter  ? 

Ihl  =  lOQkg 

73.8>  -  73.8*« 
l«b»-  lOOO'^K 
1—-0.001M 


of 


TEACHERS     EDITION. 


79 


2.   If  a 

man  buys  half 

a  ton 

of  potatoes  for  $20,  and  retails 

them   all, 

without   waste, 

at  5 

cents    a 

kilogram,   what 

profit 

does  he  make  on  the  whole  ? 

10.05 

500 

$25.00 

20. 

$5. 

3.  What  is  the  weight  of  water 
required  to  fill  a  vat  98°™  long, 
71cm  ^i(je,  and  38°°»  deep  ? 

98««»  6958i°°> 

71cm  38 


686 


6958i<"" 


55664 

20874 

264.404«'°' 
264.4041'g. 


4.   If  the  vat  of  the  last  ex- 
ample  were    filled    with    brine 
weighing  1. 04^^8  10  the  liter,  what 
would  be  the  weight  of  the  brine  ? 
264.404^^8 
1.04 


1057616 
264401 

274.981^8 


5.  If  the  vat  of  Example  3 
were  filled  with  wine  weighing 
0.981''8  to  the  liter,  what  would 
be  its  weight? 


264.404^^8 
0.981 

264404 
2115232 
2379636 


259.38^^8 


6,   What  is  the  total  weight  of 
13  men  averaging  73.48''8  each  ? 
73.48^8 
13 


22044 
7348 

955.24''8 

7.   How  many  kilograms,  and 
how  many  tons,  would  3.61 75'=^™ 
of  brick  weigh,  at  2  tons  to  a 
cubic  meter  ?  at  2.34  tons  ? 
3.6175  3.6175 

2*  2.34* 


7.235* 
=  72351^8 


144700 
108525 
72350 

8.46495* 
=  8464.951^8 


8.  From  a  barrel  containing 
67''8  of  granulated  sugar  were 
taken  three  parcels  of  2.75''8 
each,  and  four  parcels  of  7.50^^8 
each.  How  much  is  left  in  the 
barrel  ? 

2.75^8  7.5^8 

3  4 


8.25^8 


30.0^8 


80                                                ARITHMETIC. 

30.0kK 

24 

8.25 

325)7800 

38.25''« 

650 

67'"5 
38.25 

1300 
1300 

2'^  7~/'^^ 

10.  A  mass  of  21.8«  is  divided 

into  60  pills.    What  is  the  weight 

9.   Into    how    many  pills    of 

of  each  pill  ? 

325'nK  each  can  a  mass  of  7.88 

6)2180.000»8 

be  divided  ? 

363.333°»8 

Exercise  VIII. 

1.  If  a  stone  weighs  1.3^8  in  air  and  0.68''8  in  water,  and  the  stone 
and  a  block  of  wood  together  weigh  1.55''8  in  air  and  0.63"^  in  water, 
what  is  the  specific  gravity  of  the  block  of  wood  ? 

1. 55118  _  1.3kg  ^  o.25''«,  the  weight  of  the  wood  in  the  air. 

1.55kg  _  o.es^s  =  0.92''8,  the  weight  of  water  displaced  by  the  stone 
and  the  wood. 

1.3kg  _o.68''8  =  0.62''8,  the  weight  of  water  displaced  by  the  stone 
alone. 

Therefore  0.92''8  -  0.62''8  =  0.3''8,  the  weight  of  water  displaced  by 
the  wood. 

And  0.25  -4-  0.3  =  0.833,  the  specific  gravity  of  the  wood. 

2.  What  is  the  weight  of  8.17"  of  alcohol,  specific  gravity  0.83  ? 

8.17»>i  =  817''8 
0.83 
2451 
6536 
678.11*8 

8.  What  will  97' alcohol  weigh,  of  specific  gravity  0.817?  of  speci- 
fic gravity  0.819?  of  specific  gravity  0.823?  0.838?  0.847? 

0.817  0.819  0.823  0.838  0.847 

97*8  97*8  97*8  97*8  97''« 


5719 
7353 

5733 
7371 
7'.»'143*8 

r.7<;i 
7107 
79.831*8 

rtXi\i\ 

5920 

7r.2:^ 

7',t  L'l '■""-' 

81.286*8 

s 

TEACHERS     EDITION, 


81 


4,  A  bar  of  aluminum  113"™  long,  17™™  wide,  and  13™™  thick,  is 
said  to  bo  of  specific  gravity  2.57.  What  does  it  weigh  ?  If  it  really 
is  of  specific  gravity  2.67,  what  does  it  weigh  ? 

113™™  19211™™  24.9738  24.9738 

17mm  13mm  2.57  2.67 


791 
113 

5763 
1921 

174811 

124865 
49946 

174811 
149838 

19211™™ 

249  73«™™ 
=  24.973«c™ 
=  24.9738 

49946 

64.188 

66.677918 
=  66.688 

5.  What  would  be  the  specific 
gravity  of  the   bar  of  the   last 
example  if  it  weighed  65.1378? 
9fi08 


24973)65137.000 
49946 


151910 
149838 

207200 
199784 


6.  What  is  the  weight  of  a 
bar  of  aluminum  371™™  by  63™™ 
by  84™™,  specific  gravity  being 
2.63? 


371™™ 

1963332°™™ 

63™™ 

weigh 

1113 

1.963332'^e 

2226 

2.63 

233731™™ 

5889996 

84 

11779992 

93492 

3926664 

186984 

5.163563161^8 

IQCQQQOfimm 

=  5.1636^^8 

7.  An  irregular  mass  of  cop- 
per, gently  lowered  into  a  pail 
brimful  of  water,  caused  1.374^  to 
run  over.  What  did  it  weigh  if 
of  specific  gravity  8.91  ?  if  8.89  ? 
1.374>'8  1.3741^8 

8.91  8.89 


1374 
12366 
10992 

12.242^^8 


12366 
10992 
10992 

12.21486''8 
12.215'^8 


8.  What  was  the  specific  grav- 
ity of  that  copper  if  the  mass 
weighed  12.30161^8? 

8.953 
1374)12301.600 
10992 


13096 
12366 


7300 
6870 

4300 
4122 


82 


ARITHMETIC. 


9.  A  plate  of  iron  137*™  long, 
64.3<""  wide,  and  4.31«»  thick, 
weighs  277.54''8.  What  is  its  spe- 
cific gravity  ?  What  would  the 
same  mass  weigh  at  specific  grav- 
ity 7.47  ?  at  7.79  ? 

137*'""  8809.1i«" 

64.3C""  4.31 


411 

548 
822 

8809.1 


88091 
264273 
352364 

37967.221«'«' 
37.9? . 

7.309 


3797)  27754.000 
26579 


11750 
11391 

35900 
34173 

37.96722^8 
7.47 

265770547 
151868884 
265770547 

283.615''<i 

37.967221^* 
7.79 

341704989 
265770547 
265770547 

295.76 1(55 1 59*« 
295.765"*. 


10.  What  is  the  specific  grav- 
ity of  sea-water  when  a  hekto- 
liter  weighs  102.58''8  ?  what  when 
3»  weigh  30778? 

100)102.58 
1.0258 
3)3.077 
1.0257 

11.  What  is  the  specific  grav- 
ity of  a  substance  of  which  7.3'*°» 
weighs  3 1.5«? 

4.315 

73)315.000 
292 

230 
219 

110 
73_ 

370 

265 

12.  If  a  cubic  meter  of  sand 
weighs  1723"*,  what  is  its  spe- 
cific gravity  ?  If  3.4'''""  of  gravel 
weigh  7.134  tons,  what  is  the 
specific  gravity  ? 

1000)1723. 

1.723 
2.098 
34)71.340 
68 

334 
306 

280 
272 


TEACHERS     EDITION. 


83 


13.  If  a  cubic  centimeter  of 
metal  weighs  7.3«,  what  is  its 
specific  gravity  ? 

1)7.3 
7.3 

14.  What  is  the  specific  grav- 
ity of  a  fluid  weighing  2.317''s 
to  a  liter  ? 

1)2.317 
2.317 

15.  If  a  body  weigh  3.7P8  in 
air  and  2.38''8  in  water,  what  is 
its  specific  gravity  ? 


3.71 

2.789 

2.38 
1.33 

133)371.000 
266 

1050 

931 
1190 

1064 

1260 

1197 

16.  A  piece  of  ore  weighing 
3.77^2  weighs  in  water  only 
2.53 ''K.  What  is  its  specific 
gravity  ? 


3.77 
2.53 

1.24 


3.04 


124)377.00 
372 

500 
496 


17.   How  many  cubic  centi- 
meters  in   a  stone  which   loses 


17.88  of  its  weight  when  weighed 
in  water?  What  is  its  specific 
gravity  if  weighed  in  air  it 
weighs  33.78? 

1)17.8 

lygccm 

1.893 


178)337.000 
178 

1590 
1424 


1660 
1602 


580 
534 

18.  In  a  wrought-iron  bottle 
I  find  2.63^  of  quicksilver,  weigh- 
ing 35.8P8;  in  another  2.59\ 
weighing  35.193''8;  in  a  third, 
2.617\  weighing  35.57P8.  What 
is  the  specific  gravity  of  each? 
What  would  be  the  specific  grav- 
ity if  the  three  were  emptied  into 
one  vessel  and  mixed? 
13.616 


263)3581.000 
263 
951 

789 

1620 
1578 


420 
263 

1570 
1578 


84 

ARITHMETIC. 

13.588 

13,616 

259)3519.300 

13.588 

259 

13.592 

929 

3)40.796 

777 

13.59867 

1520 

=  13.599.  Am. 

1295 

2250 

2054 

19.  A  plate  of  iron  89<'«  by 

1960 

170m  ]jy  7cm  weighs  79.43k«.  What 

2054 

is  its  specific  gravity  ? 

89cm 

13.592 

17cm 

2617)35571.000 

623 

2617 

89 

9401 

151 3v» 

7851 

7cm 

15500 

1059 1*"" 

13085 

24150 

7.5 

23553 

10591)79430.0 

5970 

74137 

5234 

EXERCI 

52930 
SE  IX. 

1.  If  three  men  eat  8^8  a  week,  how  much  would  one  man  eat  at 
the  same  rate?  How  much  would  seven  men?  At  the  same  rate, 
Ijow  much  do  the  three  men  eat  in  one  day?  and  how  much  each 
man  ?  At  the  same  rate,  how  much  would  seven  men  eat  each  day  ? 
eacli  week  ?  in  five  weeks  ? 

2.67         7)8.00       3)1.14 
7  1.1 4^«        OM^ 


3)8.00 

2.67''« 


0.38 

8 


2.67 
7 


18.67 
5 


18.67'" 


2.67*«     18.67*«      93.33^* 


TEACHERS     EDITION. 


85 


2.  At  the  same  rate,  how  much 
Tvould  17  men  eat  in  3  weeks  and 
4  days  ? 

7 
3 

21 
_4 

25 
_17 

175 
25 

425 

0.38»^g 

3400 
1275 


161.501^8 


3.  If  one  hektoliter  of  oats 
is  enough  for  5  horses  1  week, 
how  much  is  enough  for  1  horse 
1  week  ?  for  1  horse  7  weeks  ? 
for  11  horses  17  weeks? 


5)1.0 
0.2^1 


201 


201 
7 

1401 


17 
17 

187 


187 
201 

37401 


4.    If  two  hektoliters  of  grain 
are  enough  for  3  horses  5  days, 


how  much  is  enough  for  3  horses 
1  day  ?  for  1  horse  1  day  ?  for 
7  horses  6  days  ? 


3 

5 

115 
5 


5)2.0 
0.4"  =  401 


3)40 


13.33 

42 

2666 
5332 


559.861  =  5601 


5.  Mix  17  liters  of  vinegar, 
costing  6  cents  a  liter,  with  39i  at 
5  cents,  2li  at  7  cents,  and  13i  of 
water  costing  nothing.  Find  the 
amount  of  the  mixture,  and  its 
cost. 


171 

391 

21' 

0.06 

$0.05 

$0.07 

1.02 

$1.95 

$1.47 

17 

i 

Pl.02 

39 

1.95 

21 

1.47 

13 

c 

^  A    A  A 

$4.44 


901 


6.  For  how  much  a  liter  must 
I  sell  that  mixture,  in  order  to 
gain  96  cents?  for  how  much  to 
clear  $1.41? 


86 


ARITHMETIC. 


14.44 
0.96 

$5.40 

14.44 
1.41 


15.85  540 


$0.06 

90)$o.40 

540 

$0,065 
90)5.850 
540^ 
450 
450 


7.  A  grocer  sold  421  kegs  of 
butter  for  $4995.25;  56  kegs 
brought  $12.50  a  keg ;  91  brought 
$11.75  a  keg;  and  100  kegs 
brought  $  12.25  a  keg.  For  how 
much  a  keg  were  the  other  kegs 
Bold? 


$12.50 
56 

7500 
6250 

$700.00 

$12.25 
100 


$1225.00 


$11.75 
91 

1175 
10575 

$1069.25 

700.00 

1225.00 

$2994.25 


$4995.25 
2994.25 

$2001.00 


66 

91 

100 

247 


421 

247 

174 


$11.50 

174)$  2001.00 
174 

261 
174 

870 
870 


8.   If  3  tons  of  coal  cost  $  15.75, 
how  many  tons  will  $36.75  buy? 

3)$  15.75  7  tons. 

$5.25  525)3675 

3675 


9.   If  5'"  of  cloth  cost  $18.75, 
what  should  7°"  cost  ? 

5)$  18.75 

$3.75 

7 


$26.25 


10.  If  a  tap  running  3.5>  a 
minute  fills  a  tub  in  16  minutes, 
how  long  should  a  tap  delivering 
5'  a  minute  be  in  filling  the 
same  tub  ? 

16 
3.5 

80 
48 

5)56.0  min. 
11.2  min. 


TEACHERS     EDITION. 


87 


11.  If  both  taps  of  the  last  ex- 
ample be  opened  at  once,  how 
soon  will  they  fill  the  tub  ? 


^ 

4.2 

2.5 

3.5 

6.6 

18 

336 
42 

18)45.6 

5. 
8.5 

85)560.0 
510 

36 
96 

500 

75.6 
30 

90 

12.  If  3  men  can  dig  378"°  of 
ditch  in  2  days,  how  long  will  it 
take  5  men,  at  the  same  rate,  to 
dig  787^ 


13. 

48^  one  tap  is  delivering  water 
at  the  rate  of  3.7'  a  minute ;  while 
out  of  it,  by  another  tap,  the 
water  is  running  at  2.5'  a  minute. 
How  long  will  it  take  to  fill  the 
tub,  beginning  with  it  empty  ? 


3.71 
2.51 

1.21 


40  min. 
12)480 


14.  A  tap  discharges  into  a 
tub  4.2'  a  minute ;  from  the  tub 
water  is  also  running,  by  a  sec- 
ond tap;   the  water  in  the  tub 


gains  30'  in  18  minutes.      How 
fast  is  the  second  tap  discharging  ? 


45.6 


^m? 

2.5 

15.   If  a  wheel 
how  often  will  it 
one  kilometer  ? 

3.1416 
1.2™ 

62832 
31416 

is  1.2'"  across, 
turn  in  going 

378°» 
189'» 

315)787.0 
630 

1570 
tub  which  will  hold 

265 

63'" 
5 

315™ 

377)100000 
754 

2460 

3.76992°* 

=  3.77'" 
=  0.00377k'" 

2262 

Into  a 

1980 

1885 

16.  How  many  times  in  a 
minute  does  the  wheel  of  the  last 
example  turn,  when  the  carriage 
is  driven  M"""  an  hour  ? 

14  ^  60  =  0.23 

265 
0.23 

795 
530 

60.95 
=  61  times. 


AEITHMETIC. 


17.  What  is  the  weight  of  the 
water  in  a  tank  if  it  would  take 
a  flow  of  8.7^  a  minute  1  hour  and 
38  minutes  to  empty  it? 

60  min. 
38  rain. 

98  min. 
8.? 

686 

784 

'  852.6» 
-  852.6»« 


18.  Replace  the  bulk  of  water 
with  oil  worth  $18.75  a  hekto- 
liter,  and  what  will  the  contents 
of  the  tank  be  worth  ? 

852.6^8  =  8.526" 
118.75 

42630 
59682 
68208 
8526 


1159.86 


Exercise  X. 

1.  A  train  leaves  Paris  at  11  o'clock  a.m.,  and  reaches  Lyons  at 
10  o'clock  P.M.  How  many  meters  does  it  travel  in  a  hour,  the  dis- 
tance from  Paris  to  Lyons  being  512.7'"°? 

There  are  11  hours  between  11  a.m.  and  10  p.m.  Therefore  the 
train  runs  512.7"™  -5- 11  =  46.609'""  =  46,609°'. 

2.  A  railroad  has  a  single  track  11.450*™  long.  How  many  rails 
4.569™  in  length  did  it  require  to  lay  the  track  ? 

There  are  two  lines  of  rails.     Therefore  length  of  rails  is 
2  X  11.450kn»  =  22.900^  =  22.900™. 
Therefore  the  number  of  rails  is  22,900  +  4.569. 

5012 
4569)22900000 
22845 


5500 
4569 

9310 
9138 


number  of  rails  required  was  5013.  Ans. 


teachers'  edition.  89 

3.  A  book  is  2.1<"^  in  thickness  ;  each  leaf  is  0.05™""  thick.     Find 
the  number  of  pages  in  the  book. 

The  number  of  leaves  is  21  ^  0.05  -  420. 
The  number  of  pages  is  2  X  420  =  840. 

4.  The  cost  of  opening  a  canal  amounts  to  $25,400  a  kilometer. 
How  much  would  a  canal  cost  which  was  113. 253'"'"  long  ? 

If   it  cost  ?  25,400  to   open    V^,   to   open  113.253  it  will  cost 
113.253  X  $25,400. 

113.253 

$25400 
45301200 
566265 
226506 


.2,876,626.200 


5.   The  expense  of  laying  out  a  paved  road  is  $12,500  a  kilometer. 
How  much  would  a  road  cost  which  was  72,053^™  long? 

If  it  cost  $12,500  to  lay  out  1^"^,  to  lay  out  72.0531''"  it  will  cost 
72.053  X  $12,500. 

72.053 
$12500 


36026500 
144106 
72053 
$900,662.50 


6.  The  cost  of  building  a  railroad  is  about  $78,000  a  kilometer  in 
France,  and  only  $25,000  in  the  United  States.  How  much  would 
it  cost  in  each  country  to  make  a  road  295.67P™  long? 

If  it  cost  $78,000  to  build  1"^,  to  build  295.671''""  it  will  cost 
295.671  X  $78,000  =  $23,062,338.  If  it  cost  $25,000  to  build  P'", 
to  build  295.67P'"  it  will  cost  295.671  X  $  25,000  =  $7,391,775. 

295.671  4)29567100 

$78000  $7,391,775 

2365368000 
2069697 


$23,062,338,000 


90  AEITHMETIC. 


7.  If  you  must  go  up  211  steps  to  reach  the  top  of  a  tower,  and 
each  step  is  195"^  high,  what  is  the  height  of  the  tower  ? 

195inin  =  0.195™. 

If  one  step  is  0.195"  high,  211  steps  are  211  X  0.195"  high. 

0.195" 

211 

195 
196 
390 


41.145" 


8.  A  house  has  5  stories,  each  story  has  19  stairs,  each  stair  is 
16""  in  height.  Calculate  how  high  the  floor  of  the  fifth  story  is 
from  the  ground. 

16«"  =  0.16". 

If  one  step  is  0.16"  high,  19  steps  are  19  X  0.16"  =  3.04",  and  4 
flights  of  19  steps  are  4  x  3.04"  =  12.16". 

0.16« 
19 

144 

16 

3.04 

4 

12.16 


9.  A  ream  of  paper  contains  20  quires,  each  quire  has  24  sheets, 
the  ream  is  13.5""  in  thickness.    Find  the  thickness  of  each  sheet. 

In  one  ream  there  are  20  x  24  =  480  sheets.     If  480  sheets  are 
13.5«»  thick,  13.5«"  +  480  =  0.028«",  thickness  of  one  sheet. 

0.028 


48)1.350 
96 
390 
384 


teachers'  edition.  91 

10.  The  equator  on  a  terrestrial  globe  measures  0.80™  in  circumfer- 
ence. By  the  aid  of  a  tape-measure  we  find  that  the  distance  between 
two  cities  on  this  globe  is  0.046™.  What  is  really  the  distance  in  kilo- 
meters between  the  two  cities?  (The  earth's  equator  is  40,075.45'^™.) 
The  ratio  of  distance  on  globe  between  the  two  cities  to  the  equator 
is  0.046™  -V-  0.80™  =  0.0575.  Therefore  the  actual  distance  between 
the  two  cities  is  0.0575  x  40,075.45^™  =  2304.338''™. 

8)0.4600  40075.451^™ 

0.0575  0.0575 

20037725 
28052815 
20037725 


2304.338'^™ 

11.  Upon  a  military  map  we  find  that  the  distance  from  Paris  to 
St.  Denis  is  78™™.  What  is  the  distance  in  kilometers  from  Paris  to 
St.  Denis?  The  map  is  made  on  the  scale  of  1  to  80,000™ ;  that  is, 
1™  on  the  map  represents  80,000™  of  actual  measurement  upon  the 
ground. 

The  actual  distance  is  80,000  times  the  distance  on  the  map ;  that 
is,  80,000  X  78™™  =  6,240,000™™  =  6.241^™. 

12.  Give  the  number  of  revolutions  made  by  the  wheels  of  a  car- 
riage in  travelling  82^^™.     The  wheels  are  1354™™  in  diameter. 

82km  ^  82,000,000™™. 

The  circumference  of  the  wheels  is  3.1416  X  1354™™  =  4253.7264™™. 
The  number  of  revolutions  is  the  total  distance  divided  by  the  cir- 
cumference of  the  wheel,  or  82,000,000™™  ^4253. 7264™™  =  19,277  times. 
3.1416  19277 

1354™™  42537264)820000000000 

125664  42537264 

157080  394627360 

94248  382835376 

31416  117919840 

85074528 


4253.7264" 


328453120 
29776084<^ 
306922720 
297760848 


92  ARITHMETIC. 


13.  IIow  many  hektars  in  a  square  kilometer  ?  how  many  ars  ? 

how  many  square  meters  ? 

Iqkm  =  100''*, 

=  10000* 

=  1000000<i«. 

14.  France  has  about  542,000<i'"».     How  many  hektars  does  it 
measure  ? 

iqkm  ^  100*'». 

.-.  5420001'^  =  542000  x  100»'*, 
=  54200000»>*. 

15.  A  piece  of  land  1224.5™  square  is  sold  at  $140  a  hektar.  How 
much  does  the  land  bring  ? 

1224.5  149.94 

1224.5  1140 

61225  599760 

48980  14994 

^^^^^  $20,991.60 

24490 

12245 


1499400.251™ 
=  149.94"*. 

16.  The  total  surface  measurement  of  the  glass  in  the  windows  of 
a  house  is  182*i"».  How  many  panes  of  53«™  by  48"^  will  it  take  to 
supply  the  windows  ? 


182i'»  =  l,820,000<»« 
53«"» 


715.4 


48°»  2544)1820000.0 

m  ^7808 

212  3920 

2544 


2544qom^  area  of  one  pane 


13760 
12720 

10400 
10176 


.'.  it  will  take  716  panes. 


teachers'  edition.  93 

17.   How  many  square  slabs  of  marble  ISO^o"^  on  the  surface  will 
it  require  to  pave  a  court  whose  area  is  25.35^™  ? 
25.351"'  =  253,500<i<=°^.' 

The  number  of  slabs  required  is  253,500i«'°  ^  ISO^^^"  -  1690. 

1690 


15) 25350 
15 
103 

90 
~135 

135 

18.  A  speculator  bought  31.0728i^a  of  land  for  $1296  a  hektar. 
For  how  much  a  square  meter  must  he  sell  it  to  realize  a  profit  of 
$1937? 

31.0728  $0,136 

^1296  310728)  $42207. 349 

310728 


1864368 

2796552 

621456 

310728 

P 40,270.3488  cost. 

1937 

profit. 

1113454 
932184 

1812709 


$42,207.3488  selling  price. 

19.   A  man  is  offered  $6000  for  2.5  ars  of  land.     He  declines  to 
sell  ;  and  soon  after,  the  town  gives  him  $25.20  a  square  meter. 
How  much  did  he  make  by  refusing  the  first  offer  ? 
2.5''  =  250^m_ 
$25.20 
250 


126000 
5040 
$6300.00 
6000 
$300 


'94  ARITHMETIC. 


20.  A  man  surveys  a  piece  of  land  and  finds  that  it  measures 
14.0715''*.     He  afterwards  discovers  that  his  chain  was  too  short  by 
0.03™.     How  can  he  calculate  the  real  superfifcial  measurement  of  his 
land  without  surveying  it  again  ?     (A  surveyor's  chain  is  10™  long.) 
10.00  -  0.03  =  9.97. 
9.97  ^  10  =  0.997. 
0.997  14.0715»'» 

0.997  0.994009 

6979  1266435 

8973  562860 

8973  1266435 


0.994009  1266435 

13.987'>» 

21.  The  railroad  from  Paris  to  Orleans  has  a  double  track ;  each 
rail  is  4™  long,  and  the  distance  from  Paris  to  Orleans  is  121'"".  What 
is  the  number  of  rails  used  in  laying  the  track  ?  The  width  of  the 
road  is  15™ ;  how  many  hektars  of  land  does  the  road  include  ? 

There  are  four  lines  of  rails.  4  X  121''™  =  484"™  =  484,000™  of  rails. 
If  one  rail  is  4™  long,  in  484,000™  there  are  484,000^-4  =  121,000  rails. 
15™  =  O.OIS''™.     The  area  of  road  is 

121"™  X  0.015"™  =  1.815^"™  =  181.5''^    Ans. 
121"™  4)484000  121^» 

_J  121000  rails.  ^j^ 

484km  605 

121 
1.8151^ 

22.  Calculate  the  number  of  ars  in  a  surface  which  a  ream  of  paper 
(180  sheets)  will  cover.     The  sheets  are  30.3"™  long  and  195™™  wide. 

1{)5"""  =  19.5«™. 

The  area  of  one  sheet  is  30.3°™  X  19.5°™  =.  590.85<»°™. 

The  area  of  480  sheets  is  480  X  590.85'«°»  =  283,608«»««»  =  0.283608*. 

19.5  590.85 

30.3  480 

~585  4726800 

585  236340 


590.85<>«»  283608.00V" 


teachers'  edition.  95 


23.  A  pile  of  wood  is  4^.25^  long,  1.33"^  thick,  and  2.60«»  high. 
How  many  sters  are  there  in  it? 

1^  =  1<'^'".     In  the  pile  of  wood  there  are 

4.25  X  1.33  X  2.60  =  14.6965«l""=14.696^   Ans. 
4.25  5.6525 

1.33  2^ 

1275  339150 

1275  113050 

425  14.69650c^°» 

5.6525 

24.  A  beam  is  7.070™  long ;  its  two  other  dimensions  are  0.258"^ 
and  87*"™.     Find  its  volume. 

87mm  =  0.087°'.  Its  volume  is  0.258"»  x  0.087™  X  7.070™  =  0.15869«^™. 

0.258  0.022446 

0.087  7.07 

1806  157122 

2064  157122 


0.022446  0.15869322'=^'™ 

25.    A  bar  of  iron  3™  long  measures  45™™  square  on  the  end  where 
it  has  been  evenly  cut.     The  bar  is  heated  and  drawn  out  to  a  greater 
length  by  being  passed  through  an  orifice  24™™  square.     What  is  the 
length  of  the  bar  after  the  operation  ? 
45™™  =  0.045™.     24™™  =  0.024™. 

The  volume   of  the   bar   is   0.045™  x  0.045™  X  3™  =  0.006075«bm. 
The  area  of  the  end,  after  the  bar  has  been  boated  is 

0.024™  X  0.024™  =  0.0005 7()i™. 
Therefore  the  length  of  the  bar  is  0.006075  -h  0.000576  =  10.547™. 

0.024™  10.547™ 

0.024 


576)6075.000 


96  576 

48 


0.002025*1™  0.0005761^ 


3150 
2880 

3  2700 

0.006075°^™  2304 

3960 
3892 


96  ARITHMETIC. 


26.  A  reservoir  is  1.50™  wide,  2.80™  long,  and  1.25™  deep.  Find 
how  many  liters  it  contains  when  full,  and  to  what  height  it  would 
be  necessary  to  raise  it  that  it  might  contain  lO**^™, 

The  volume  of  the  reservoir  is  1.5  x  2.8  X  1.25  =  5.25«^™  =  5250». 
The  area  of  the  bottom  is  1.5  X  2.8  =  4.2<i™ ;  therefore,  in  order  to 
contain  lO^bm^  the  height  must  be  lOc*"™  -j-  4.2i™  =  2.38™. 

1.5™  2.38™ 

^  42)100.00 

120  84 

?^  160 

4.201™  126 

2100  336 

840 
420 


5.2500 


27.  Suppose  a  box  to  be  3.75™  long,  3.50™  wide,  and  0.50™  high. 
How  much  lime  would  it  take  to  fill  it  with  mortar,  reckoning  that 
Icbm  of  ijjne  after  being  slaked  becomes  1  80°''™  of  mortar  ? 

Volume  of  the  box  is  3.75  x  3.50  x  0.50  =  6.5625«b™.  Since  l"*""  of 
mortar  when  slaked  becomes  1.8°''™,  the  box  will  hold  6.5625«'»™  of 
slaked  mortar,  which  is  the  same  as  6.5625"''™  -^  1.8  =  3.646«''™  ( f 
dry  mortar. 

3.75  3.646<=''™ 

_^  18)65.  eis*"™ 

1875  54 

1125  116 


13.125  10^ 
05  82 

6.5625«'»™  2^ 

105 

28.  A  chest  has  the  following  dimenoions :  1.17™,  00"',  1.04"'. 
IIow  many  cakes  of  soap  13«™  square  on  the  bottom  and  29°™  tliick 
could  bo  put  in  it?  0.12  of  the  volume  of  the  chest  must  be  deducted 
for  packing. 


teachers'  edition.  97 

The  volume  of  a  cake   of  soap   is   13  x  13  X  29  =  490 1««™      The 
volume  of  the  chest,  deducting  waste  of  room  in  packing,  is 

1.17  X  0.90  X  1.04  X  0.88  =  0.9637056«bm  =  963705.6««'". 
Therefore  the  chest  will  hold  (963,705.6  -=-  4901)  cakes  of  soap. 


13 

1.17 

1.04 

468 

117 

12168 

0.9 

1.09512 

0.88 

196 

13 
39 

4901)963705.6 
4901 

13 

169 

47360 
44109 

29 
1521 

32515 
29406 

338 

4901ccm 

876096 
876096 
0.963  7056«''m 

29.  A  cubic  meter  of  dry  plaster  makes  I.IS^'^"  when  tempered; 
tempered  plaster  increases  1  in  every  100,  twenty-four  hours  after 
it  is  mixed.  What  volume  of  tempered  plaster  would  be  obtained 
from  55  sacks  of  25^  each  of  dry  plaster? 

25i  =  0.025«i'«".  The  volume  of  the  plaster  is  55x0.025°bm=i.375cbin^ 
As  l«^«i  makes  1.18c»»n  when  tempered,  1.375<=*>°^  will  make  1.375 
X  1.18«^°»  =  1.6225<=bra.  In  twenty-four  hours  its  volume  will  be 
1.01  X  1.6225«^"i  =  1.6387«b"\ 

0  025^^™  1.375cbm  1.6225 

55  _L1^  101 

—  11000  16225 

lor                                    1^'^^                                16225 
±:5_  1375  


1.375«^'"  1.62250«b'«  1.6387"''™ 

30.   A  reservoir  is  2.80™  long,  1.50™  wide,  and  1.25™  deep.     How 
many  liters  will  be  required  to  fill  0.80  of  it? 

1.5™  4.201™  52501 

2.8  JL25  0.8 

120  ^^^^  4200.0 

30  840 

420 

4.201™  5.2500<=b™ 

=  5250\  volume. 


98  ARITHMETIC. 


31.  A  man  buys  1415"  of  wheat  for  $  3.50  a  hektoliter ;  but  the 
measure  used  proves  too  small,  the  mistake  amounting  to  3*  in  every 
hektoliter.  What  wa.s  the  quantity  of  wheat  delivered  to  the  pur- 
chaser, the  cost,  and  the  reduction  which  ought  to  be  made  to  him 
on  account  of  the  error? 

The  mistake  was  3^  in  100',  or  he  received  only  0.97  of  1415" 
=  1372.55".  If  1"  of  wheat  cost  $3.50,  1415"  cost  1415x|3.50 
=  $4952.50.  A  reduction  of  0.03  of  $4952.50  =  $  148.58  ought  to  be 
made. 

1415"  1415  $4952.50 

0.97  $3.50  0.03 

9905  70750  $148.5750 

12735  4245 


1372.55"  $4952.50 

32.  The  dimensions  of  a  tile  are  as  follows :  length  22*"",  width 
11""",  thickness  55"'"'.  Find  the  volume  of  the  tile,  and  the  number 
of  tiles  in  a  pile  of  25<='"". 

55min  ^  5  5cm.  The  volume  of  a  tile  is  22  x  11  X  5.5  =  1331«^". 
25obm  =  25,000,000«'"».  In  the  pile  there  will  be  25,000,000  h-  1331 
=  18,732  tiles. 

22«'"  18732 

y  1331)25000000 

22  1331 


22 


11690 


242  10648 


55  10420 

1210  9317 

1210  11030 


1331.0ocm  10648 


3820 
2662 


33.  The  measurement  of  a  pile  of  wood  shows  that  a  ster  could  be 
filled  from  it  25.08  times.  Give  the  volume  of  the  pile  in  cubic 
meters,  reckoning  the  length  of  the  logs  to  be  1.15". 


teachers'  edition.  99 

The  volume  of  a  pile  is  1  X  1  X  1.15  x  25.68  -  29.532<=bm.    Ans. 
25.68 

l_15cbin 


12840 

2568  , 

2568 

29.5320«to°» 

34.  A  liter  of  air  weighs  1.273s.     How  much  does  a  cubic  meter 
of  air  weigh  ? 

icbm  =  looQi.     Therefore  l^^™  of  air  weighs  1000  x  1.273s  =  1273^ 
=  1.2731^8.   Ans. 

35.  A  package  of  candles  which  weighs  465^  is  sold  at  28  cents. 
What  is  the  price  of  a  kilogram  of  candles  ? 

IK  of  candles  costs  i^O.28  ^  465  =  $0.000602.     Therefore  l^s  costs 
1000x^0.000602  =  $0,602. 

36.  How  many  times  would  3.243'  of  water  611  a  liter? 

As  1'  of  water  will  fill  a  cubic  meter,  3.243'  will  fill  3.243«^'n 
=  32431.     3243  times.   Ans. 

37.  Give  the  weight  in  kilograms  of  43.4578«°™  of  pure  water. 
43.4578<=°°»  of  water  weigh  43.4578s  =  0.0434578''«. 

38.  The  volume   of  an   engine's   axletree  is  0.245'='^'".     Find  its 
weight,  the  specific  gravity  of  the  iron  being  7.8. 

0.245c*"n  of  water  weigh  0.245*,  and  0.245°^'"  of  iron  weigh 
7.8x0.245^=1.911*. 
0.245 
7.8 


1960 
1715 

1.9110* 


39.  Calculate  the  volume  of  a  gram  of  the  following  substances : 
proof  spirit,  specific  gravity  0.865;  tin,  specific  gravity  7.291 ;  lead, 
specific  gravity  11.35;  copper,  specific  gravity  8.85;  silver,  specific 
gravity  10.47;  cork,  specific  gravity  0.240. 


100 

ARITHMETIC. 

The  volume  equal 
vided  by  the  specific 

8  1«"»,  which,  filled  wi 
gravity. 

th  water,  weighs  1«,  di- 

(i) 

(lii.) 
0.088«'» 

(V.) 

0.095«« 

865)  1000.00 
865 

1135)  1( 

)0.000 
9080 

1047)  100.000 
9423 

1350 

865 

9200 
9080 

5770 
5235 

4850 

(vi.) 

(iv.) 

0.113<«» 

885)  100.000 

4  lOyccm 

(ii.) 

0.14ccm 

24)  100.000 
96 

7291)1000.00 

- 

885 

40 

7291 

27090 

1150 
885 

24 
160 

2650 

144 
160 

40.  Olive  oil  costs  60  cents  a 
kilogram.     Wiiat  is  the  price  of  a 
liter?     The  specific  gravity   of 
olive  oil  is  0.914. 

As  1^1  costs  $1.87,  r  costs  0.792 
X$  1.87  =  $1.48. 

$1.87 

0.792 

V  of  olive  oil  weighs  0.914*8. 
As  !"«?  costs  .f  0.60,  V  costs  0.914 
X  10.60  =  $0,548. 

374 
1683 
1309 

0.914 
$0.60 

$0.54840 


41.  Pure  alcohol  copts  $  1 .87  a 
kilogram.  What  is  the  price  of 
a  liter?  The  specific  gravity  of 
alcohol  is  0.792. 

1»  of  alcohol  weighs  0.792>«. 


$1.48 

42.  A  man  wants  to  build  a 
shed  large  enough  to  hold  135»* 
of  wood ;  if  the  shed  is  to  be 
3™  high  and  5"  wide,  how  long 
must  it  be  ? 

135-»  -  135«t»'>».  The  ground 
area  is  3x5  =  15*»"».  Therefore 
the  height  must  be  135  +15  =  7™. 


TEACHERS     EDITION. 


101 


43.  In  a  country  where  fire- 
wood is  cut  LIS'"  long  what 
height  must  the  sides  of  the  ster 
be  to  hold  a  cubic  meter  ? 

Tlie  height  must  be 

Icbrn^ll^qm^  0.86207"". 

0.86207'" 
116)100  00000 
928 

720 
696 

240 
232 


800 


44.  If  a  ster  of  cork  cost 
120.00,  how  much  would  lOOi^s 
cost,  the  cork  weighing  one 
quarter  as  much  as  water? 

l«t  of  cork  weighs  250^s,  and 
costs    $20.00.      lOO'^g  will    cost 

100 

2:50' 

=  $8.00. 


45.  A  liter  of  powder  weighs 
825«.  What  would  be  the  vol- 
ume of  a  charge  for  a  gun  if  the 
charge  weighed  5*??  Calculate 
the  volume  in  cubic  centimeters. 

The  specific  gravity  of  powder 
is  0.825.  It  takes  (1  ^  0.825)««'" 
of  powder  to  weigh  1^ ;  therefore 
to  weigh  58  it  takes  5'^'^  ^  0.825 


6.06 


825)5000.00 
4950 


5000 
4950 


46.  Out  of  gold  which  weighs 
19.362  times  as  much  as  water, 
sheets  of  gold  foil  are  made 
which  are  O.OIO"*"*  in  thickness. 
What  surface  would  3^  of  gold 
cover? 

0.010»^°»  =  O.OOlc"^.  The  vol- 
ume of  the  gold  is  3°«™  -^  19.362 
=  0.154943"°'".  Therefore,  the 
surface  is  0.154943<'«'"  --  0.001°'" 
=  154.943i«'>\ 

0.154943 


19362)3000.000000 
19362 


106380 
96810 


95700 
77448 

182520 
174258 

82620 
77448 

51720 


47.  Find  the  weight  of  an  oak 
board  3.25"!  long,  0.31  "Mvide,  and 
0.04™  thick  ;  the  specific  gravity 
of  the  oak  being  0.808. 


102 


ARITHMETIC. 


The  volume  of  the  board  is 
3.25  X  0.31  X  0.04  =  0.0403«»'«n. 
P*""  of  oak  weighs  0.808* ;  there- 
fore 0.0403«''"»  weigh  00.403 
X  0.808*= 0.0325624«  =  32.5624^8. 

3.25 

0.31 

325 
976 


1.0075 
0.04 


0.040;  lOO^"™,  volume. 

0.0403 
0.808 


3224 
3224 

0.0325624 


48.  Find  the  weight  of  a  bar 
of  iron  having  the  following 
dimensions:  length  3.0",  width 
6''"»,  thickness  2*^;  the  specific 
gravity  of  the  iron  being  7.8. 
3.6'"  =  360^™. 

360 
6 

2160 
2 
4320ccm^  volume. 

4320 
7.8 


34560 
3024 

33696.0* 
«=  33.696k« 


49.  How  many  lead  balls  each  weighing  27«  could  be  obtained  by 
melting  a  mass  of  lead,  cubic  in  form,  the  edge  measuring  0.356",  the 
specific  gravity  of  the  lead  being  11.35? 

0.356™  =  35.6«°». 


35.6<«»» 

45118.016 

18966 

35.6 

11.35 

27)5120S9 

2136 

225590080 

27 

1780 

135354048 

242 

1068 

45118016 

216 

1267.36 
35.6 

45118016 
512089.481601 

260 

OA9 

760416 
633680 
380208 


45118.016<«»,  volume. 


178 
162 

169 
162 


teachers'  edition.  ■  103 

50.  Marble  costs  1 30.95  a  cubic  meter,  and  the  specific  gravity  of 
marble  is  2.73.  If  a  block  of  marble  weighs  1260'^^^  what  is  its  vol- 
ume, and  what  is  it  worth  ? 

I'^to'"  of  marble  weighs  2.73'.     1260'^^  =  1.26'. 

0.45i5cbm  0.4615 


273)  126.0000  ^  30.95 

1092  23075 

1680  41535 

1638  13845 


420  114.28 

273 

1470 
1365 

51.   Sea-water  contains  28  parts,  by  weight,  of  salt  in  1000.     A 
liter  of  sea-water  weighs  1.025^s.    How  many  kilograms  of  salt  could 
be  obtained  from  126.276842'=»"»  of  sea-water  ? 
V^s  of  sea-water  contains  0.028^8  of  salt. 

126.276842  129433.753 

1.025''8  0.028''8 


631384210  1035470024 

252553684  258867506 

^^^^^^^^^  3624.145084'^g 


129.433753050''g 

52.  An  empty  cask  weighs  17.06'^K;  when  filled  with  water  it 
weighs  275.8''8.  How  many  liters  does  it  hold?  How  many  casks 
of  this  size  would  it  require  to  receive  the  wine  from  a  vat  containing 
3.008°^'^  ? 

The  cask  will  hold  275.8'^s  -  17.06''«  =  258.741^8  of  water.  It  takes 
258.741  of  water  to  weigh  258.74i^g.  Therefore  the  cask  will  hold 
258.741. 

3.008<'i>°'  =  30081.  If  one  barrel  holds  258.741,  to  hold  30081,  it  will 
take  3008  -  258.74  =  12  barrels. 

275. 801^8  ^2 

17.06  25874)300800 

25874 


258.74i'8 


42060 


104 


ARITHMETIC. 


53.  It  takes  about  204.8'  of  wheat  to  sow  a  hektar.  How  many 
cubic  meters  would  it  take  to  sow  a  square  kilometer  ? 

iqkm  =  looha.  P*  will  require  100  x  204.8»  =  20,480'  =  20.48«»>'».  Aru. 

54.  A  piece  of  road  l"""  long  and  7"*  wide  is  to  be  macadamized ; 
the  macadamizing  is  to  be  33«"  deep  ;  it  costs  43  cents  a  cubic  meter 
to  prepare  the  stones.     What  will  enough  for  the  road  cost? 

Itm  =  IQQQm  .    330m  =  O.SSn*. 

0.33  2310 

7  $0.43 

2.31  6930 

1000  9240 

2310.00  $993.30 

55.  A  gasometer  holds  28,000*5*°  of  gas.  How  many  jets  would 
this  gasometer  feed,  when  each  jet  burns  125'  an  hour,  and  is  used 
4  hours  every  evening? 

Each  jet  will  burn  4  X  125'  =  500'  each  evening.  28,000«»>°» 
=  28,000,000'.   The  gasometer  will  feed  28,000,000  ^  500  =  56,000  jets. 

56.  The  city  of  Venice  is  situated  in  the  midst  of  a  great  lake  of 
salt  water,  communicating  with  the  sea,  and  all  the  rain- water  is 
caught  for  the  cisterns.  Ordinary  years  the  fall  of  rain  in  Venice 
is  82*=™ ;  the  surface  of  the  city,  after  the  canals  have  been  deducted, 
is  520''*;  reckoning  the  population  at  115,330,  how  many  liters  a 
day  of  rain-water  could  each  inhabitant  have? 

520'"'  =  5,200,000<»°» ;  82<=™  =  0.82'". 

The  average  amount  of  rain-water  is  5,200.000  X  0.82  =  4,264,000«'»«» 
=  4,264,000,000'.  Each  person  can  use  per  year  4,261,000,000 
+  115,530,  or,  per  day,  4,264,000,000  +  (115,530  X  365)  =  101.118'. 


0.82 

115,530 
365 

1U1.11»' 

5200000 

4216845)426400000.000 

16400000 

577650 

4216845 

410 

693180 

4715r)00 

4264000.00 

346590 

4216845 

42168450 

4986550 
4216845 
7697050 
42I6.S45 
34802050 
33734760 

teachers'  edition.  105 

57.  Find  the  weight  of  a  bar  of  iron  5.35"  long,  4.56°'"  thick,  and 
3.51"^™  wide.  Find,  also,  the  width  of  an  oak  beam  4.30°*  long, 
9.12*=°^  thick,  which  has  the  same  weight.  The  specific  gravity  of 
the  oak  to  be  reckoned  at  1.026,  that  of  the  iron  7.788. 

5.35m  _  535cm^  4  3om  _  430cm_  The  volume  of  the  oak  beam  is 
67,258.5969928  ^  1.0262  =  65,554.1 88«=°\  The  area  of  one  side  of  the 
oak  beam  is  430  X  9.12  =  3921.6i<"" ;  therefore  thickness  is  65,554.188<=«'" 
-j-3921.6^°i"  =  16.72°'". 

4.56<='^  16.1424  8636.184 

3.54  535  7.7888 


1824 

807120 

69089472 

2280 

484272 

69089472 

1368 

807120 
8636.1840<"''», 

,  volume. 

60453288 

16  1424q°°» 

60453288 

65554.188 

67258.596992s 
=  67.259'^g. 

1026)67258596.992 

6156 

16.72<'«^ 

6698 

392: 

16)655541.88 

5130 

5685 
5130 
5559 

39216 

263381 
235296 

51.30 

280858 

4296 

274512 

4104 
1929 

63468 

1026 

9039 

8208 

8212 

58.  Give  the  specific  gravity  and  volume  of  a  body  weighing  35'^8 
in  air  and  30^^  in  water. 

The  weight  of  water  displaced  by  the  body  is  5^^. 

The  weight  of  body  in  air  is  35^8. 

Therefore  specific  gravity  is  35  -5-  5  =  7.  35^  of  water  weigh  35^^^ ; 
35  -T-  7  =  5\  volume. 


106  ARITHMETIC. 


69.  A  Bter  of  piled  oak  wood  weighs  42o*« ;  the  specific  gravity 
of  the  wood  is  0,74.  What  is  the  volume  occupied  by  the  spaces 
between  the  logs  ?  For  how  much  must  lOO^s  of  separate  sticks  be 
sold  in  order  to  bring  the  same  amount  as  when  sold  by  the  ster ; 
aster  selling  for  $2.20? 

If  there  were  no  spaces  between  the  logs,  the  ster  of  wood  would 
weigh  740''K.  Therefore  the  spaces,  if  filled  with  wood,  would  weigh 
740^8  _  425»'8  =  SIS"*.  Therefore  volume  of  spaces  is  315  ^  740 
=  0.42568<""».  lOQi^K  ought  to  be  sold  for  |f f  of  $  2.20  =  $  220  h-  425 
=  $0,518. 

0.425680"°'  $0,518 

74  Jsi. 50000  425)  $220,000 
296  2125 

190  ^ 

148  425 


420 

370  3250 

500 

444 

560 

60.  Wrought  iron  sells  for  $7.00  per  lOO'^R.  A  bar  of  iron  4.5o°» 
wide,  3.3°™  thick  costs  $5.08;  what  is  its  length,  reckoning  the  spe- 
cific gravity  of  the  iron  at  7.4  ? 

$7.00  per  lOO''*  is  the  same  as  $0.07  per  kilogram.  An  iron  bar 
that  costs  $5.08  must  weigh  5.08  ^  0.07  =  72.57143*8,  and  its  volume 
is  72.57143  -i-  7.4  =  9.8066»  =  9806.5''cm.  The  area  of  an  end  of  the 
bar  is  4.5«™  X  3.3<""  =  14.85vm  Therefore  the  length  is  9806.6 
+  14.85  =  660.4''™  =  6.604°*. 

9.8066>  660.4°» 

74)725.7143  1485)980650.0 

666  8910 

f7  •  -^ 

^4  ^ 

444  5500 

l03 
666 


TEACHERS     EDITION. 


107 


61.  Experiment  shows  that 
water  weighs  770  times  as  much 
as  air  ;  and  the  specific  gravity 
of  mercury,  in  comparison  with 
water,  is  13.6.  How  many  liters 
of  air  will  it  take  to  weigh  as 
much  as  a  liter  of  mercury  ? 

"Water  is  770  times  as  heavy  as 
air,  and  mercury  is  13.6  times 
as  heavy  as  water.  Therefore 
mercury  is  13.6  X  770  times  as 
heavy  as  air. 


13. 


770 


ters,  the  surface  which  can  be 
covered  by  the  sheets  thus  ob- 
tained. The  specific  gravity  of 
the  lead  is  11.3. 

The  volume  of  the  lead  is  753 
H- 11.3  =  66.637'  =  0.066637<"'°». 

0  imm  _  0.0001™.     The  surface 
of  the  lead  is 

0.066637'='""  H-  0.0001  °» =  666.37i"*. 
66.63?  ^'^s- 

113)  7530.000 
678 
750 
678 
720 
678 
420 
339 
810 
797 


9520 
9520 
10472.0 

62.  A  mass  of  lead  weighing 
7531^8  is  made  into  sheets  O.l''^"* 
thick.     Calculate,  in  square  me- 

63.  A  rectangular  sheet  of  tin  of  uniform  thickness  is  85*'™  wide, 
1.35'"  long;  it  weighs  268«.  What  is  its  thickness,  reckoning  the 
specific  gravity  of  tin  at  7.3  ? 

The  volume  of  the  tin  is  268 -=- 7.3  =  36.7109<=<'«' ;  1.35°»  =  135<'°'. 
The  area  of  the  tin  is  135<=™  X  85""=  11,475<*<'™  ;  therefore  its  thick- 
ness is  36.7109«"»  -^  ll,475«i<""  =  0.0032«°'. 


36.7109««°' 

135 

0.0032^ 

73)  2680.0000 

85 

11475)36.7109 

219 

675 
1080 

34325 

490 

23859 

438 
520 

11475qcin 

22950 

511 

90 

73 
700 

657 

108 


ARITHMETIC. 


64.  The  fine  coal  which  collects  about  the  shafts  of  the  mines  and 
in  the  coal-yards,  was  for  a  long  time  wasted,  because  it  could  not 
be  burned  in  stoves  and  grates.  Now,  this  dust  is  mixed  with  tar  in 
proportion  of  92''k  of  dust  and  8^^  of  tar ;  the  mixture  is  heated,  and 
afterwards  pressed  in  rectangular  moulds  of  14. 75*™,  18.5<=°»,  and  29<'™ ; 
each  one  of  these  blocks  weighs  lO''^ ;  they  are  sold  at  $3.00  a  ton, 
and  make  excellent  fuel  for  heating  steam  boilers.  Give  the  specific 
gravity  of  this  fuel ;  also,  the  sum  which  would  be  realized  in  thus 
utilizing  800,000*  of  coal  dust,  the  cost  of  tar,  mixing,  etc.,  being 
f  0.50  a  ton. 

Volume  of  a  block  is  14.75  x  18.5  x  29  =  7913.305<^  =  7.9133051. 
Specific  gravity  is  10  -r-  7.913305  =  1.264.  800,000*  of  coal  dust  will 
make  800,000'  ^  0.92  =  869,565.217*  of  the  mixture.  869,565.217*  at 
$2.50  per  ton  =  869,565.217  X  $2.50  =  $2,173,913.04. 


14.75«° 
18.5 

7375 
11800 
1475 

272.875 
29 

2455805 
545750 

7913.305~» 

1.264 
79133) 100000.000 
79133 


208670 
158266 

504040 
474798 

292420 


869565.217 
92)  80000000.00 
736 
640 
552 
880 
828 
520 
460 
600 
552 
480 
460 
200 
184 
160 
92 
680 
644 

869565.217 

$2.50 
43478260850 
1739130434 
$2,173,913.04 


TEACHERS     EDITION. 


109 


65.  A.  bar  of  iron  a  millimeter 
square  on  the  end  will  break  un- 
der a  tension  of  SO'^s,  Find  the 
length  at  which  a  suspended  bar 
of  iron  will  break  from  its  own 
weight,  the  specific  gravity  of 
the  iron  being  7.8. 

301^8  =  0.03*. 

The  volume  of  the  iron  bar  is 
0.03  ^  7.8  =  0.00384615'^t.in  The 
area   of   an   end   of  the    bar  is 

1mm  y^   1mm  ^   ^qram  ^  O-OOOOOl^l™. 

Therefore  the  length  of  the  bar 
is  0.00384615«^«^  -^  O.OOOOOli'" 
=  3846.15">. 

0.00384615°^'^ 
78)0.30000000 
234 


660 
624 


360 
312 

480 
468 

120 
78 

420 
390 

66.  Fifty-three  kilograms  of 
starch  are  obtained  from  lOO'^s 
of  wheat.  A  hektar  of  land  pro- 
duces 1363  of  wheat;  a  hekto- 
liter  of  wheat  weighs  78^8.  If 
the  wheat  harvested  from  a  field 
measuring  2*'*  and  33i»»  is  taken 


to  a  starch   factory,  how   much 
starch  will  be  made  from  it? 

0.53'^s  of  starch  are  obtained 
from  Ps  of  wheat.  1^  of  wheat 
weighs  O.VS^K.  P^produces  1363 
X  0.78^^  of  wheat  =  1063.14i^«, 
2ha  33qm  _  2.0033iia.  2.0033^* 
produce  2.0033  x  1063.1 4"^^  = 
2l29.7883f>2kg  of  wheat.  The 
amount  of  otarch  is  1128.7878'^k. 
1363 
0.781's 


10904 
9041 

10G.;.14'^8 
2.0033 
318912 
318942 
212628 
2129.788362»'e 

2129.788 
0.53 
6389364 
10648940 

1128.78764''K 

67.  A  gardener  wishes  to  pro- 
vide glass  for  his  hot-beds.  The 
beds  cover  2.65*  ;  the  panes  will 
cover  0.75  of  the  whole  surface, 
the  rest  being  taken  up  by  the 
frames  and  alleys.  First,  find 
how  many  panes  measuring  45<=™ 
by  37°™  it  will  take  to  cover  the 
beds ;  then  find  the  price  of  the 
glass,  at  a  cost  of  95  cents  a 
square  meter. 


110  ARITHMETIC. 


45««  =  0.45°' ;  37"°  =  0.37™ ;  2.65»  =  265«i'°. 

Total  area  of  the  glass  is  0.75  of  265'«'°  =  198.75«»"».  The  area  of 
one  pane  is  0.45  X  0.37  =  0.1665*i™.  Therefore  the  number  of  panes 
needed  is  198.75  +  0.1665  =  1194.  At  |0.95  per  square  meter, 
198.75'»'°  will  cost  198.75  X  $0.95  =  $  188.81. 


0.45"' 

1194 

198.75 
$0.95 

99375 

178875 

0.37" 

315 
135 

1665)1987500 
1665 

8225 
1665 

15600 
14985 

0.16651" 

$188.81 

6150 


68.  A  jar  full  of  water  weighs  1.325*«;  filled  with  mercury  it 
weighs  12.540''8,  What  is  the  capacity  of  the  jar,  and  its  weight? 
The  specific  gravity  of  the  mercury  is  13.59. 

The  weight  of  the  jar  and  the  jar  full  of  mercury  is  12.540''-. 
The  weight  of  the  jar  and  the  jar  full  of  water  is  1.325''8.  Therefore 
the  difference  in  weight  between  the  mercury  and  the  water  is 
12.540>'8  -  1.325''8  =  11.215''8.  13.59  -  1  =  12.59,  the  specific  gravity 
of  a  liquid  of  which  the  jar  full  without  the  jar  weighs  11.215''«. 
Hence  the  capacity  of  the  jar  is  11.215  -f- 12.59  =  0.890791.  0.89079> 
of  water  weigh  0.890791^8.  Therefore  weight  of  jar  is  1.325  -  0.89079 
-  0.43421''K  =  434.21*. 


12.540k« 
1.325 

1259) 

0.890791 
1121.50000 
10072 

1.325001^3 

0.89079 

11.215k« 

0.4312i*« 

11430 
11331 

9900 
8813 

10870 
10072 

TEACTTERS'    EDITION.  HI 

69.  A  hektoliter  of  rape-seed  weighs  GoI^k,  and  32^  of  oil  can  be 
extracted  from  it.  How  many  kilograms  of  the  seed  will  it  take  to 
make  a  hektoliter  of  oil  ? 

Ihi  =.  lOQi.  If  32^  of  oil  can  be  extracted  from  eS^«  of  seed,  V  of  oil 
can  be  extracted  from  63  -4-  32  =  1.96875'^«  of  seed,  and  100'  of  oil  can 
be  extracted  from  100  X  1.96875'^g  =  196.875''s  of  seed. 

1.96875 


32)63.00000 
32 
310 
288 
220 
192 
280 
256 
240 
224 
160 
160 


70.  Common  burning  gas  is  0.97  of  the  weight  of  air,  and  a  liter 
of  air  weighs  1.293s.  In  a  shop  there  are  65  jets,  each  one  of  which 
burns  123'  an  hour,  and  is  used  5  hours  in  the  winter  evenings. 
Calculate  the  weight  of  the  gas  used  in  a  month,  and  the  expense 
of  lighting  the  shop,  when  gas  costs  6  cents  a  cubic  meter. 

1'  of  gas  weighs  0.97  X  1.2938  =  1.25421«.  65  jets,  each  burning  1 23' 
an  hour,  and  used  5  hours  an  evening  for  30  days,  will  use  65  X  5  x  30 
X  123' =  1,199,250',  the  weight  of  which  is  1,199,250  x  1.25421k 
=  1,504,111.348  =1504.11134'^«.  1,199,250' =  1199.25«'»n.  The  ex- 
pense at  1 0.06  per  cubic  meter  is  1199.25  x  |0.06  =  $71.96. 


1.2938 

123' 

1199250 

1199.25 

0.97 

65 

1.254218 

$0.06 

9051 

615 
738 

1199250 
239850 

$71.9550 
=  $71.96 

79951 

479700 

L.254218 

5 

599625 

39975' 

239850 

30 

119925 
1504111.342508 

1199250 

112  ARITHMETIC. 


71.  A  merchant  buys  one  kind  of  wine  at  30  cents  a  liter,  another 
kind  at  21  cents  a  lit«r ;  he  mixes  the  two  kinds  by  putting  5'  of  the 
first  with  8^  of  the  second.  For  how  much  a  liter  must  he  sell  the 
mixture  in  order  to  gain  $3.75  a  hektoliter? 

5>  at  $0.30  per  liter  cost  $1.50. 

81  at  $0.21  per  liter  cost  $1.68. 

Therefore  13»  of  the  mixture  cost  $1.50 +  $1.68  =  $3.18,  and  I'costa 
$3.18  ^  13  =  $0.2446.  Again,  if  $3.75  per  hektoliter  is  equivalent 
to  a  gain  of  $0.0375  per  liter,  to  make  $3.75  per  hektoliter  the  mer- 
chant must  sell  the  wine  for  $0.0375  +  $0.2446  =  $0.2821  per  liter. 

$0.30                              $0.21  0.2446 

5  ?  13)3.1800 


$1.50  11-68 


26 


1.50  58 


$3.18  52_ 

GO 
52 

80 


72.  If  it  requires  360  tiles  to  drain  an  ar  of  land,  what  will  it  cost 
to  drain  17.784''*,  when  the  tiles  cost  $20  a  thousand,  and  the  ex- 
pense of  laying  is  the  same  as  the  cost  of  the  tiles? 

The  expense  of  laying  the  tiles  and  their  cost  is  $40  per  thou- 
sand. 17  784»"'  =  1778.4*.  To  drain  1778.4*  of  land  1778.4x360 
tiles  =  640,224  tiles  =  640.224  thousand  are  needed.  640.224  thou- 
sand at  $40  per  thousand  cost  640.224  X  $40=  $25,608.96.   Ans. 

1778.4  640.224 

360  $40 

1067040  $25608.960 

53352 


640224.0 


73.  It  is  found  in  building  that  hewn  stone  of  medium  durability 
ought  not  to  support,  as  a  permanent  weight,  more  than  0.07  of  the 


teachers'  edition.  113 

weight  that  it  would  require  to  crush  it.  A  certain  kind  of  stone 
used  for  building  will  be  crushed  under  a  weight  of  250^^  a  square 
centimeter.  What  is  the  greatest  height  to  which  a  wall  constructed 
of  this  material  can  be  safely  carried,  the  specific  gravity  of  the  stone 
being  2.1  ? 

250^^^  per  square  centimeter  is  equivalent  to  250,000^  per  square 
centimeter.  0.07  of  250,000s  =  17,5008  ought  to  be  the  pressure  on  a 
square  centimeter.  Therefore  volume  of  imaginary  prism  ought  to 
be  17,500  ^  2.1  =  8333.33««'",  or  the  height  ought  to  be  8333.33'='^ 
=  83.333°^.  8333.33«<=°» 

21)175000.00 
168 

"to 

63 

"To 

74.  Several  kinds  of  wines  are  mixed  as  follows :  245^  at  20  cents 
a  liter,  547'  at  15  cents  a  liter,  344'  at  25  cents  a  liter.  How  much 
does  the  mixture  cost  a  liter  ? 

2451  at  $0.20  per  liter  cost  $49.00 
547'  at  $0.15  per  liter  cost  $82.05 
344'  at  $0.25  per  liter  cost  $86.00 


1136' of  the 

mixture  cost 

$217.05 

'herefore 
245 

$0.20 

1»  costs  $217.05- 

547 

$0.15 

-1136  =  $0,191. 
344 

$0.25 

Ans. 
1136) 

$0,191 
217.050 

$49.00 

2735 
547 

$82.05 

1720 
688    ■ 

1136 

10345 

$86.00 

10224 

1210 
1136 

75.  A  farmer  wishes  to  drain  a  field  of  8.75'*'^.  Each  hektar  re- 
quires 750""  of  ditches.  The  opening  of  these  ditches  costs  10  cents  a 
running  meter ;  the  tiles  are  30*=™  long,  and  cost  $15  a  thousand.  He 
pays  2  cents  a  meter  for  laying  the  tiles,  and  4  cents  a  meter  for  fill- 
ing the  ditches.     What  is  the  cost  of  draining  the  field? 


114 


AKITIIMETIC. 


There  are  required  8.75  x  750"*  =  6562.5°»  of  ditches.  The  expense 
of  opening  the  ditches,  laying  the  tiles,  and  filling  the  ditches  is 
$0.10  +  .$0.02  +  $0.04  =  $0.16  per  meter.  6562.5°>  will  cost  6562.5 
X  $0.16  =  $  1050.00.  SO^-"  =  O.S".  For  6562.5'°,  6562.5  -f-  0.3  =  21,875 
tiles  are  necessary.  The  tiles  cost  $15  per  thousand.  Therefore 
21.875  thousand  cost  21.875  x  $15  ==  $328.13.  Hence  cost  of  draining 
the  field  is  $1050.00  +$328.13  =$1378.13. 


8.75 

6562.5 

21.875 

$1050.00 

750» 

$0.16 

$15 

328.13 

43750 

393750 

109375 

$1378.13 

6125 

(K5fi25 

21875 

6562.50°» 

$1050.000 

$328,125  = 

'  $328.13. 

Exercise  XI. 

1.   Find  the  prime  ftictors  of  148 ;  264;  178;  183;  173;  187;  346 
343. 

2^1148  2»[264  2[178  3|183 

37  3  [^  89  61 

22  X  37.  Ans.  11  2  x  89.  Ans.  3  X  61.  Ans. 

23x3x11.   Ans. 


1|173 
173 
1x173.  Ans. 


11 1 187 
17 
11  X  17.  Ans. 


2 1 346 
173 
2x173.  Ans. 


7' 1 343 
1 
7'.  Ans. 


2.  Find  the  prime  factors  of  210 ;  353;  5280;  231;  31,416;  1369; 
1368. 


210 

105 
35 
7 
2x3x5x7.  Am. 

2» 


353 
1x353.  Ans. 


5280 


165 
_55 
11 
2'»X 3x5x11.  Ans 


31231 
71  77 

n 

3x7x11.  Ans. 


31416 


3927 


1309 


3711369 
37 
37x37.  Ans. 


1368 


171 


187 


19 
2»x3»xl9.  Ans. 


17 


2'x3x7xn  xl7.  Ans. 


TEACHERS     EDITION. 


115 


3.   Find  the  prime  factors  of  247 ;  327 ;  179 ;  83 ;  2125 ;  2353  ;  2333. 

13 1 247  3 1 327  1  [179  1^83 


19  109 

13  X  19.  A71S.       3  X  109.  Ans. 


179 
1x179.  Ans. 


83 
1  X  83.  Ans. 


5^12125 
17 
5^  X  17.  Ans. 


13 1 2353 
181 
13x181.  Ans. 


1 1 2333 
2333 
1x2333.  Ans. 


4.    Find  the  prime  factors  of  165  ;  168  ;  2148  ;  16,662  ;  321  ;   1551  ; 


38. 


31165 

5155 

11 


168 
21 

7 


2212148 

3  I   537 

179 


2116662 
31  8331 


3x5x11.  Ans.  2^  X  3  X  7.  Ans.  2^  x  3  X  179.  Ans.  2  x  3  x  2777.  Ans. 


3 1 321 
107 
3  X  107.  Alls. 


311551 
111   517 
47 
3  X  11  X  47.  Ans. 


2[33 
19 

2  X  19.  Ans. 


5.   Find  the  prime  factors  of  82  ;  129 ;  72 ;  66 ;  68  ;  65 ;  76 ;  86  ; 
3;  142. 

2j66  22  [68 

3133  17 

11        22  X  17.  Ans. 
X32.  Ans.     2x3x11.  Ans. 


2[82 

3|129               23  72 

41 

43               32    9 

2  X  41.  Ans. 

3  X  43.  Ans.               1 

2^x32.  A 

5[65 

22[76               2186 

13 

19                   43 

23 1 


11 


2 1 142 
7L 


5x13.  ^ns.     22x19.  ^ns.    2x43.  ^Ins.    23x11.  ^ns.    2  X  71.  ^ns. 

6.   Find  the  prime  factors  of  326 ;  368  ;  464 ;  292 ;  362 ;  365 ;  730  ; 
42. 


21326 

2*1 368 

2* 

1464 

22 1 292 

163 

23 

29 

73 

2  X  163.  Ans. 

2*x23.  Ans. 

2*x29.  Ans. 

22  X  73.  Ans 

2|362 

5|365 

2 

730 

2 

142 

181 

73 

5 

365 

3 

21 

2  X  181.  Ans. 

5  X  73.  Ans. 

73 

7 

2  X  5  X  73.  Ans.    2x3x7.  Ans. 


116 


ARITHMETIC. 


7.   Find  the  prime  factors  of  868 ;  999 ;  822 ;  1346 ;  7641 ;  6234 
234. 

2^1868  3«|999  21822  2|1346 

7  [217  37  SgU  673 

31  3'x37.   ^715.  137  2x673.  Ans. 

2''  X  7  X  31.  Ans.  2  x  3  x  137.  Ana. 


3^17641 
283 
33x283.  Ans. 


6234 


3117 
1039 
2x3x1039.  Ans. 


2    234 

32  [117 
13 
2x3^x13.  Am. 


8.   Find  the  prime  factors  of  579;  577;   212;   126;   128;  8192; 
8190. 

3 1 579  1|577  2^1212  2  1126 

193  577  53  32  [^ 

3x193.  Am.     1x577.  Am.       2^x53.  Am.  7 

2x32x7.  Am. 


271128 

21' 18192 

2 

8190 

1 

1 

32 

4095 

21  Ans. 

2".  Am. 

5 

455 

7 

91 

13 
2x32x5x7x13.  Am. 


9.  Find  the  prime  factors  of  8197;  3125;   2401;   1331;    78,309; 
25.179. 


7|8197 
1171 
7x1171.  Am. 

32 

S'^jSl 

5^  A 
78309 

25 

1 
m. 

Am. 

7* 
7*. 

2401 

1 
Ans. 

3x7 

3 

7 

11 

1P|1331 

1 

IP.  Am. 

25179 

7 

8701 

8393 

11 

1243 

1199 

32x7x 

113 
11x113. 

X^ 

109 
11X109.  Am. 

TEACHEES     EDITION. 


117 


Exercise  XII. 

1.   Find  the  prime  factors  of  8.4 ;  7.6;  1.08;  0.144;  0.036;  0.037; 
21.45. 

8.4  =  84x0.1.    7.6  =  76x0.1.    1.08  =  108x0.01.    0.144  =  144x0.001. 
22 184  22  [76  22M08  2^  144 

3  m  19  32 1^  32 9 

7  1  1 

22x3x7x0.1.^ns.  22xl9x0.1.^ns.  22x32x0.01.^ws.  2*x 32x0.001. ^?7s. 


0.036  =  36  X  0.001.  0.037  =  37  X  0.001. 
22 1 36  1[37 

3^\_9  37 

1  1x37x0.001.  Ans. 
22x32x0.001.  Ans. 


21.45  =  2145x0.01. 


2145 


715 
148 


13 


3x5x11x13x0.01.  Ans. 


2.    Find  the  prime  factors  of  14.6;   2.61;   21.2;    78.54;   0.5236; 
0.00052. 


14.6  =  146x0.1         2.61  =  261x0.01. 

21.2  =  212x0.1. 

2|146                          32 1 261 

22 1 21 2 

73                                  29 

53 

2  X  73  X  0.1.  Ans.       32  x  29  x  0.01.  Ans. 

22x53x0.1.  An 

78.54  =  7854  x  0.01.                       0.5236 

=  5236x0.0001. 

2 

7854 

22 
7 
11 

5236 

3 

3927 

1309 

187 

1309 

7 

187 

11 

17 

17                        22x7xllx 

17x0.0001.  Ans. 

2x3x7x11x17x0.01.  Ans. 

0.00052  =  52x0.00001 

22  [52 

13 

2^X13X0.00001.  Ans. 

118 


ARITHMETIC. 


3.   Find  the  prime  factors  of  86.7 ;  48.3 ;  99.99 ;  5.04 ;  1.485 ;  0.216. 
86.7  =  867x0.1.  48.3  =  483x0.1.  99.99  =  9999x0.01. 


3 
IT' 


867 

289 

1 


31483 

71161 

23 


32 
11 


9999 


UU 


101 


3x172x0.1.  ^ns.    3  X  7  X  23  X  0.1.  ^ns.     3^x11  X  101  x  0.01.  ^rw. 
5.04  =  504x0.01.       1.485  =  1485x0.001.      0.216  =  216x0.001. 


231504 

3^1  63 

7 


1485 
55 


11 


216 

27 

1 


2^x32x7x0.01.  Ans.    33x5x11x0.001.  Ans.    23x3^x0.001.  Am. 


4.   Find  the  prime  factors  of  34.87  ;  32.4;  5.115;  71.2;  2.993. 
34.87  =  3487x0.01.  32.4  =  324x0.1.  5.115  =  5115x0.001. 


11  [3487 
317 
11x317x0.01.  Ans. 


22 1 324 
3*1  81 
1 
22  X  3*  X  0.1.  Alls. 


5115 
.  1 705 


341 
31 


71.2  =  712x0.1. 
23  [712 
89 
23x89x0.1.  Am. 


3x5x11x31x0.001.  Am. 

2.993  =  2993x0.001. 
41 1 2993 
73 
41x73x0.001.  Am. 


Exercise  XIII. 


1.  Find  the  G.  C.  M.  of  27  and  33. 

3[27 33 

9      11 
3.  Am. 

2.  Find  the  G. CM.  of  13  and  39. 

13[13 39 

1        3 
13.  Am. 


3.  Find  the  G.C.M.  of  8  and  28. 

2*  1 8 2^ 

2        7 
2^  =  4.  Am. 

4.  Find  the  G.  C.  M.  of  27  and  45. 

3«[27 45 

3        6 
3«  =  9.  Am, 


TEACHERS     EDITION. 


119 


5.   Find  the  G.  CM.  of  81  and 
108. 

3^181     108 

3        4 

33=27.  Ans. 


12. 


6.  Find  the  G.C.M.  of  4,  10, 

2[4  10  12 
2  5  6 
2.  Ans. 

7.  Find  the  G.  C.  M.  of  4.  6. 


10. 


2|4  6  10 
2.  3  5 
2.  Ans. 

8.   Find  the  G.  C.  M.  of  9,  12, 


21. 


3|9    12    21 


3      4      7 

3.  Ans. 

9.   Find  the  G.  C.  M.  of  10,  15, 

25. 

5|10    15    25 

2      3      5 

5.  Ans. 

10.   Find  the  G.  C.  M.  of  14, 

98,  42. 

2 

14    98    42 

7 

7    49    21 

1      7      3 

2x 

.  7  =  14.  Ans. 

11.   Find  the  G.  C.  M.  of  30, 
18,  54. 

2130     18     54 
3115      9    27 
5      3      9 
2x3  =  6.  Ans. 


12.   Find  the  G.  C.  M.  of  14, 
56.  42. 


2 

14     56 

42 

7 

7    28 

21 

1       4 

2x7  =  14. 

3 

A77S. 

13.   Find  the  G.  C.  M.  of  96, 
36,  48. 


22 

96 

36     48 

3 

24 

9     12 

^X 

8 
3  = 

3      4 

12.  Ans 

14.  Find  the  G.  C.  M.  of  84, 
105,  63. 


84     105    63 


28       35     21 


3 

7^ 
^4        5      3 
3x7  =  21.  Ans. 

15.  Find  the  G.  C.  M.  of  24, 
60,  84,  128. 

2^124  60  84  128 
6  15  21  32 
22  =  4.  Ans. 

16.  Find  the  G.  C.  M.   of  45, 
81,  27,  90. 

32 1 45  81  27  90 
5  9  3  10 
32  =  9.  Ans. 

17.  Find  the  G.  C.  M.  of  78, 
18,  54.  42. 


2 

78     18     54    42 

3 

39      9     27     21 

13      3      9       7 
2x3  =  6.  Ans. 

120 


ARITHMETIC. 


18.  Find  the  G.  C.  M.  of  98, 
28.  70.  42. 


98    28     70    42 


49    14    35    21 


7      2      5      3 
2x7=14.  Ans. 

19.  Find  the  G.  C.  M.  of  96, 
112,  80,  32. 

2^196  112  80  32 
6  7  5  2 
2*  =.  16.  Ans. 

■     20.   Find  the  G.  C.  M.  of  24, 
96,  48,  120. 

2^  1 24    96    48     120 
3  I   3     12      6  "~l5 
14      2        5 
23x3  =  24.  Ana. 

21.   Find  the  G.  C.  M.  of  84, 
252,  108,  210. 


2 

SI 

252 

168 

210 

3 

42 

126 

84 

105 

7 

U 

42 

28 

35 

2        6        4        5 
2  X  3  X  7  =  42.  Ans. 


22.  Find  the  G.  C.  M.  of  33, 

88,  77,  55. 

11 1 33    88    77    55 
3      8      7      5 
11.  Ans. 

23.  Find  the  G.  C.  M.  of  252, 
315.  420.  504. 


252    315    420    504 


84     105     140     168 


12      15      20      24 
3x7  =  21.  Ans. 


24.  Find  the  G.  C.  M.  of  128, 
192,  320,  368,  432. 

2*1128     192     320    368    432 
8       12      20      23      27 
2*  =  16.  Ans. 

25.  Find  the  G.  C.  M.  of  loO, 
204,  357.  459. 

17 1 136  204  357  459 
8   12   21   27 
17.  Ans. 

26.  Find  the  G.  C.  M.  of  909, 
1414,  2323,  4242. 

1011909  1414  2323  4242 
9   14   23   42 
101.  Ans. 


Exercise  XIV. 


1.  Find  the  G.C.M.  of  2479 
and  3589. 

2479)3589(1 
2479 
lOllllO 
31   HI 

37)2479(67 
222 


37.  Ans. 


259 
259 


2.  Find. the  G.C.M.  of  3015 
and  6195. 


3045        6195 


609 


1239 


203     )    413(2 
406 

7)203(29 
14 


5x3x7  =  105.  Ans. 


(J3 
63 


TEACHERS     EDITION. 


121 


3.  Find 

the   G.C.M.   of   568 

and  712. 

8|568        712 

71   ) 

2 
32 

8.  Ans. 

89(1 
71 
18 
9 

1 

4.  Find  the  G.C.M.  of  11,023 

and  C)493. 

6493) 11023  (1 

6493 

10 

4530 

3 

453 

151)6493(43 

604 

453 

151.  A 

453 

ns. 

5.  Find  the  G.  C.  M.  of  1485 
and  2160. 


1485 


2160 


297 


432 


11  16 

5x33  =  135.  Ans. 

6.  Find  the   G.  C.  M.  of  7040 
and  7392. 

32  7040        7392 
10 


220 


231 


22 


11)231(21 
22 

Ti 

n 

32  X  U  =  352.  Ans. 


7.  Find  the   G.  C.  M.  of  2760 
and  4485. 

3 
5 


2760 

4485 

920 

1495 

184 

299 

23)299(13 
23 
~69 
()9 
3  X  5  X  23  =  345.  Ans. 


8.  Find  the   G.C.M.  of   1177 
and  2675. 

1111177 

107)2675(25 
214 
~535 
535 

107.  Ans. 


9.  Find  the  G.C.M.  of  78,473 
and  94,653. 

78473)94653(1 
78473 
10 


16180 


1618 


809)  78473  (97 
7281 
~5663 
5663 


809.  Ans. 


122 


ARITHMETIC. 


10.  Find  the  G. CM.  of  36,143 
and  10,283. 

10283)  35143  (3 
30849 
2 
19 


4294 


2147 


113)10283(91 
1017 


113.  Ans. 


11.   Find  the  G.C. 
and  61,087. 

44323)61087(1 
44323 


113 
113 

M.  of  44,323 


4 

16764 

3 

4191 

11 

1397 

127.  Ans. 


127)44323(349 
381 
622 
608 
1143 
1143 


12.  Find  the  G.  CM.  of  232,353 
and  39,699. 

11139699   232353 
9 


3609 
401     ) 


21123 
2347(5 
2005 


342 


11x9  =  99.  Ans. 


_57 

19)401(21 
38 
21 
19 
2)19(9 
18 

1 


13.   Find  the  G.  C.  M.  of  33,853 

and  35,017. 

33853)35017(1 
33853 
4 


1164 


291 

97)33853(349 
291 
475 
388 
873 
873 

97.  Am. 


14.  Find  the  G.C.M.  of  5115 
and  7254. 


3 

5115   7254 

5 
11 

1705   2418 
341 

31)2418(78 

217 
248 

248 

3x 

31  =  93.  Ana. 

15.  Find  the  G.C.M.  of  2268 
and  3348. 
4 
9 
3_ 

21     31 
4x9x3=.  108.  Am. 


2268 

3348 

567 

837 

63 

93 

TEACHERS     EDITION. 


123 


16.   Find  the  G.  C.  M.  of  1003 

and  2419. 

18.   Find  the  G.  CM.  of  30,072 
and  133,784. 

1003)2419(2 

8 
7 
3 

30072       133784 

2006 

3759         l()723 

7|  413 

59)1003(17 
59 
413 
59.  Ans.            413 

537           2389 

179)2389(13 

179 

599 

537 

2 
3 

42 
21 

17.   Find  the  G.  C.  M.  of  419 
and  52,.301. 

419)  52301  (124 
419 
1040 

7 

1|179 
179 

838 

3 

5 

2021 

1676 

345 

115 

21 

7 

1|419 
419 

19.  Fi 

and  10,8: 

9 
11 

9x4 

Qd  the  G.C.M.  of  4527 
56. 

4527          10836 

3 

7 

1.  Ans. 

473 
43 

3  =  387 

7 1 1204 
)         172(4 
172 

Ans. 

20.  Find  the  G.  C.  M.  of  17,104  and  27,794. 


17104   27794 


8552    13897 

1069)13897(13 

1069 


3207 
3207 


2  X  1069 --2t6^.  Ans, 


124 


ARITHMETIC. 


Exercise  XV. 


1.  Find  the  G.  C.  M.  of  855, 
1197,  1596. 


855 


1197 


1596 


285 


399 


95       7|13^ 

19  19 

3X19  =  57.  Ans. 


41532 

7|T33 

19 


2.   Find  the  G.  C.  M.  of  3864, 
3404,  3657. 


3864 

340  ; 

9(i6 

851 

161 

23}_ 


851 


3)3657 
1219 

1219 


37 
23.  Ans. 


53 


3.  Find  the  G.  C.  M.  of  15,561, 
11,115,  13,585. 

13111115       13585       15561 


855     5 
95    11 


1045     7 
209     9 


1197 


171 


19  19 

13x19  =  247.  Am. 


19 


4.  Find  the  G.C.M.  of  2943, 
2616,  4578. 

3  2943        2616        4578 

9 1  981       8 1 872    21 1526 

109  109    71  763 

109 

3x109-327.  Aiu. 


5.  Find  the  G.  C.  M.  of  1177. 
1391,  1819. 

11|1177 

107)1819(17 
107 
749 
749 

107)1391(13 
107 
321 
321 

107.  Ans. 

6.  Find  the  G.  C.  M.  of  4939, 
1347,  3143. 

11 14939 

449)1347(3 
1347 

449)3143(7 
3143 
449.  Am. 

7.  Find  the  G.  C.  M.  of  740, 
333,  296. 

21740        9 1 333        8|296 
10  [370  37  37 

37 
37.  Am. 

8.  Find  the  G.  C.  M.  of  833, 
1785,  1309. 

7 1 833        311785        11 1 1309 
119        5 I  595  119 


119 


119.  Am. 


TEACHERS     EDITION. 


125 


9. 

6 

Find  the  G.  C.  M 
7326                  7 
1221                 11 

.  of  735 

8547 

6,  8547,  9768,  22,755. 

.     8  9768 

11    1221 

5 
41 

22755 

11 

1221 

4551 

10 

214^ 

111 
111.  Ans 

Find  the 
)94           3 

G.C.J 
7491 

111 

.1.  of4£ 

4 

11 

94,  7491, 

9988 

2497 

111 

9988,  12,485, 
5  12485 
11     2497 
227 

111 

16,571. 

73 

1112497          11 

2497 

227) 16571 

227                 227 
227.  Ans. 

227 

1589 
681 
681 

Exercise  XVI. 


21. 


1.   Find  the  L.C.  M.  of  6,  14, 

2[6     14     21 
?      J    21 
2  X  3  X  7  =  42.  Ans. 


2.   Find  the  L.  C.  M.  of  8,  12, 
3,24. 

^    ;^    3     24. 

24.  Ans. 


3.    Find  the  L.C.  M.  of  6,  10, 


15. 


2|6     10    15 
3      ^     15 
2  X  3  X  5  =  30.  Ans. 


4.   Find  the  L.  C.  M.  of  9,  12, 

18,4. 

21^    12     18    i 
31         6      9 
2      3 
22  X  32  =  36.  Ans. 


5.   Find  the  L.  C.  M.  of  15,  21, 


35. 


3115     21     35 


J    35 
■  105.  Ans. 


24 


3x5x 

6.   Find  the  L.  C.  M.  of  12,  20, 
;?     20    24 


10     12 


6 


5  3 

23  X  3  x  5  =  120.  Ans. 

7.   Find  the  L.C.M.  of  14,  24, 

28. 

22];^    24     28 

6  7 

23  X  3  X  7  =  168.  Ans. 


20. 


8.   Find  the  L.  C.  M.  of  12,  15, 

3 1 12     15     20 
^      ^    20 
22x3x5  =  60.  Ans. 


126 


ARITHMETIC. 


32. 


77. 


\)\). 


13. 


9.  Find  the  L.  C.  M.  of  16,  24, 

2»|;^    24    32 
3      4 
2*  X  3  =  96.  Am. 

10.  FindtheL.C.M.of21,33, 

3|21     33    77 
T    W    77 
3x7x11  =  231.  Ans. 

11.  Find  the  L.  C.  M.  of  27, 33, 

3^127    3'^    99 
3  11 

33x11  =  207.  Ans. 

12.  FindtheL.C.M.  of7,  11, 
17     11     13 


7x11x13  =  1001.  Ans. 


13.   Find  the  L.C.  M.  of  77, 55, 


35. 


6|77    55    35 

77  ;;    /f 

5x7x  11  =  385.  A71S. 


14.   Find  the  L.C. M. of  16, 18, 
27,  72. 

2»|16    X^    27    72 

2  27      JJi 

2*  X  3»  -  432.  Ans. 


15. 

22,33 
2 

FindtheL.C.M.  of  10, 12 

60. 

;0    X;Z    22    33    60 

3 

;;     33    30 

22  X 

11     10 
3x5x11  =  660.  Ans. 

16.  Find  the  L.  CM.  of  15, 
16,  18,  20,  22,  24. 


2 

15 

16 

18 

20 

22 

24 

2 

15 

8 

9 

10 

11 

12 

2 

15 

4 

5) 

^ 

11 

6 

3 

15 

2 

9 

11 

3 

5      2      3  11 

2^x3^x5x11  =  7920.  Ans. 

17.  Find  the  L.  C.  M.  of  56, 
64,  70,  84,  112. 


2 

^iS    64 

70 

84 

112 

2 

32 

35 

42 

56 

22 

16 

35 

21 

28 

7 

4 

35 

21 

7 

4      5      3 

2«x  3x5x7  =  6720.  .4ns. 

18.  Find   the  L.  C.  M.  of  48, 
54,  81,  144,  162. 

2    ^^    H    H     144     162 


32 


72      81 


8        9 
2*  X  3*  =1296.  Ans. 


19.  Find  the  L.  C.  M.  of  75. 
100,  120,  150,  180. 


0 

n 

100 

120 

150 

180 

2 

10 

12 

15 

18 

3 

^ 

6 

15 

9 

2        5        3 
23  X  32  X  5»  »  1800.  Ans. 


TEACHERS     EDITION. 


127 


20.  Find  the  L.  C.  M.  of  112, 
168,  196,  224. 


2^ 

m 

168 

196 

224 

2 

42 

49 

56 

7 

21 

49 

28 

3        7        4 
25x3x72  =  4704.  An8. 

21.  Find  the  L.C.M.  of  7,  14, 
15,  21,  45. 

3|J     14    l^    21     45 
14  7    15 

2x3^x5x7  =  630.  Am. 

22.  Find   the  L.C.M.  of  16, 
25,  81. 

[16     25    81 
16x25x81  =  32,400.  Ans. 

23.  Find  the   L.  C.  M.  of  26, 
39,  52,  65. 

13|$Z^    39    52    65 
3      4      5 
22x3x5x13  =  780.  Am. 


24.  Fi 

72,  225,  ^ 
2=^ 

nd  the  L.  C.  M.    of  80 

18. 
80     72    225    48 

2 

10      9     225      6 

^      ^255       3 
2*  X  32  X  52  =  3600.  Ans. 

25.  Find   the   L.  C.  M.   of  10, 
20,  30,  40,  50,  60. 
2 
2 

5^ 

2      5      3 
23x3x52  =  600.  Am. 


X0 

%^ 

30 

40 

50 

60 

20 

25 

30 

10 

25 

15 

26.  Find  the  L.  C.  M.  of  30, 
42,  105,  70. 

2 1 30    42     105     70 


r^  n  105  0^ 

2x3x5x7  =  210.  Ans. 

27.  Find  the  L.  C.  M.   of  36 
t,  35,  20. 

22  36     24    35     20 

3 

9       6     35      ^ 

3      2    35 
23  X  32  X  5  X  7  =  2520.  Ans. 

28.  Find  the  L.  C.  M.  of  7,  11, 
14,  15. 

1/     11     14     15 

2x7x11x3x5  =  2310.  Am. 

29.  Find  the  L.C.M.   of  12, 

18,  27,  63,  28. 

2  112     18     27     63     28 
2      6      ^     27    63     14 


32    3 


27     63 


3       7 
22x33x7  =  756.  Am. 

30.  Find  the  L.  C.  M.  of  34, 

26,  Qb,  85,  51,  39. 

2  34    26     65     85    51     39 


l^    JL^    65     85    51     39 


n  i^  51  39 


17    13 
2x3x5x13x17  =  6630.  Am. 

31.  Find  the  L.C.M.   of  12, 
18.  96.  144. 


23 

n    %?>     96 

144 

2 

12 

18 

3 

6 

9 

2        3 

2^x32  =  288.  Am. 


128 


ARITHMETIC. 


32.  Find  the  L.  C.  M.  of  84, 
156.  63,  99. 


22 

84 

156  63 

99 

3 

n 

:}'J  63 

99 

3 

l.i  21 

33 

i;^     7    11 
2»x3'X  7x11  X  13=36,036.  Am. 


33.  Find  the  L.  C.  M.  of  17. 
51,  119,  210. 

17|;T    51     119    210 
3        T    210 
2x3x7x5x17  =  3570.  Am. 


34.  Find  the  L.  C.  M.  of  16, 
30,  48,  56,  72. 


2 

n  30 

48  56  72 

2» 

15 

24  28  36 

3 

15 

6   7   9 

5      2      7      3 
2*x  32x5x7  =  5040.  Am. 


36.  Find  the   L.  C.  M.  of  27, 
33,  64,  69.  132. 

2m    n    54    69     132 
31  27    69      66 

9     23      22 
2«X3>X  11x33 -27,324.  Am. 


36.  Find  the  L.  C.  M.  of  li 
26.  39,  (55,  180. 


2 

l^    26  39 

65 

180 

3 

n    39 

65 

90 

5 

n 

65 

30 

i:i      6 

2«x3«x  5x13-2340.  Am. 


44 

126 

198 

280 

330 

22 

63 

99 

140 

165 

n 

63 

99 

70 

165 

21 

33 

70 

55 

3 

38 

10 

55 

37.  Find  the  L.  C.  M.  of  44, 
126,  198,  280,  330. 

2 
2 
3 
7 
5^ 

33      2    ;; 

2»  X  3«  X  5  X  7  X  11  =  27,720.  Am. 

38.  Find  the  L.  C.  M.  of  50, 
338,  675,  975. 

5 
5 

3^ 

3.38        9      n 
2  X  3»  X  5"''  X  132  ^  228,150.  Am. 

39.  Find  the  L.C.M.  of  552, 
575,  920. 


50 

338 

675 

975 

10 

338 

135 

195 

;2 

338 

27 

39 

2» 

552  575  920 

5 

69  575  115 

2» 
2»x3x, 

69  115   ?3 

3    5 
5"  X  23 -13,800.  Am. 

40.  Find  the  L.  C.  M.  of  228. 
304,  342. 

21228    304    342 

21114     152    171 

191  g7      76    171 

4        9 

2«  X  3»  X  19  -  2736.  Am. 

41.  Find  the  L.  C.  M.  of  1080 
and  1260. 

1080         1260 


10 
2 

3« 


108 


126 


54 


63 


2»x3»x5x7-7560.  ilrw. 


TEACHERS     EDITION. 


129 


42.  Find   the  L.  C.  M.  of  600 
and  480. 


23 

600 

480 

3 

75 

60 

5 

25 

20 

25  X 

5 
3x52  = 

4 

=  2400.  Ans. 

43.  Find  the  L.  C.  M.  of  1564 
and  1032. 

22  1564        1932 
23      391           483 

17  21 

22  X 17  X  3  X  7  X  23  =  32,844.  Ans. 

44.  Find  the  L.  C.  M.  of  2530 
and  1760. 

212530        1760 


5  1 265 


880 


11     253 


176 


23  16 

25  X  5  X  11  X  23  =  40,480.  Ans. 

45.  Find  the  L.  C.  M.  of  936 
and  2925. 

321936        2925 
13  rioi  325 


8  25 

23  X  32  X  52  X  13  =  23,400.  Ans. 

50.  Find  the  L.  C. 

71539 

11     77 


46.  Find  the  L.  C.  M.  of  3432 
and  4032. 


3432 


4032 


429 


504 


143  168 

2*  X  35  X  11  X  13  =  576,576.  Ans. 

47.  Find  the  L.C.M.  of  1875 
and  2425. 

52 1 1875        2425 
75  97 

3x5^x97  =  181,875.  Ans. 


48.  Find  the  L.  C.  M.  of  1632 
and  2976. 


£3 

1632 

2976 

£2 

204 

;^,72 

3 

51 

93 

25x3x1 

49.  Fin 

and  2233. 
11 

7 

17            31 
7x31  =  50,592.  Ans. 

1  the  L.  C.  M.  of  1001 

1001        2233 
91          20:5 

13     29 
7  X  11  X  13  X  29  =  29,029.  Ans. 

M.  of  539  and  1463. 
1463 


209 


7     19 
7^x11x19  =  10,241.  Ans. 


130 


ARITHMETIC. 


Exercise  XVII. 


1.  Find   the  L.  C.  M.   of  424 

5.  Find  the  L.C.M.  of  1003 

and  583. 

and  1357. 

8|424 

53)583(11 
583 
G.  C.  M.  =  53. 
L. CM.  =  11x424  =  4664.  Ans. 

1003)1357(1 
1G03 
6\  354 

59)1003(17 
50 

2.  Find   the   L.  C.  M.   of  319 

413 

and  407. 

413 

111319        407 
29          37 

L.  C.  M.  =  17x  1357= 23.069.  Am. 

G.C.M.  =  11. 

L.C.M.  =  29x407  =  ll,803.  ^ns. 

6.  Find  the  L.  C.  M.  of  899 

3.  Find  the  L.C.M.  of  1679 
and  1932. 

and  961. 

899)961(1 

4 
3 

7 

1932 
483 
161 

23)1679(73 
161 
69 

899 
2|  62 

31)899(29 
62 
279 
279 

G.C.M.  =  23.      «? 
L.C.M.=73xl932=141,036.  Am. 

L.  C.  M.  =  29  X  961  =  27.869.  Am. 

4.  Find  the  L.  C.  MT  of  1003 
and  2419. 

7.  Find  the  L.C.M.  of  407, 

1003)2419(2 
2006 
7|413 

59)1003(17 
59 

703,  444. 

11|407 

37)703(19 
37 
333 

413 

333 

L.C.M. 

413 
-17x2419-41.123.  i4n«. 

L.C.M.-llxl9x444-92.796. 
Ans. 

TEACHERS 

EDITION.                                    Idi 

8.  Find   the    L.C.M.  of 

411, 

2  322 

959,  2055. 

7  161 

ill     ^59     2055 

23)851(37 

959)2055(2 

69 

1918 

161 

137)959(7 

161 

959 

L.  C.  M.  =  7x  2055  =  14,385.  Ans. 

23  X  3  X  7  X  23  X  37  X  53 

=  7,577,304.  Ans. 

9.  Find  the  L.C.M.  of  22 

1  and 

351. 

12.  Find  the  L.  C.  M.   of  539 

221)351(1 

and  253. 

221 

Ill  253    539 

10  130 

23      49 

13)221(17 

L.  C.  M.  =  23  X  539  =  12,397.  Ans. 

221 

L.C.M.  =  17x351  =  5967. 

Ans. 

13.  Find  the  L.  C.  M.  of  2943, 
2616,  4578. 

10.  Find  the  L.C.M.  of  1426 

and  989. 

8 1 2616 

327)2943(9 

211426 

2943 

4 
3 

276 
69 

23)713(31 

69 

23 

23 

L.C.M.=2x31x989=61,318.  Ans. 

11.  Find  the  L.  C.  M.  of  3864 

3404,  3657. 

22 

3864 

3404     3657 

3 

966 

851     3657 

23 

322 

851     1219 

14        37 


53 


2 
327 


2616     2943    45^ 


1308     2943     2289 


4  9  7 

2x4x7x9x327  =  164,808.  Ans. 

14.  Find  the  L.  C.  M.  of  2863 
and  1151. 

L.C.M.  =  1151x2863 

=  3,295,313.  Ans. 

15.  Find  the  L.  C.  M.  of  1177, 
1391,  1819. 

10711177     1391     1819 
11        13        17 


132                                              ARITHMETIC. 

1111177 

18.  Findthe  L.C.M.  of  23,309 

107)1301(13 

and  10,753. 

107 

L.  C.  M.  =  10,753  x  23,309 

321 

=  240,631,677.  Ans. 

321 

L.  C.  M.  =13x17x1177=260,117. 

19.  Find  the  L.C.M.  of  4939 

Ans. 

and  3143. 

7 1 3143 

16.  Find  the  L.  C.  M.  of  5317 
and  2863. 

7 1 2863 

409)5317(13 
409 

449)4939(11 
449 
449 
449 
L.  C.  M.  =  11x31 43  =  34,573.  ^rw. 

1227 
1227 

20.  Find  the  L.C.M.  of  4199 

L.C.M.=13x  2863-37,219.  ^ns. 

and  6137. 

13 [4199 

323)6137(19 

17.  Find  the  L.C.M.  of  12,703 

323 

an4  12,879. 

2907 

L.C.M.  =  12,703x12,879 

2907 

=  163,601,937.  Ans. 

L.C.M.  =  19x4199=79,891..1n«. 

Exercise  XVIII. 


1.  Reduce  to  whole  or  mixed 
numbers  ^. 

J^=lf.  Am. 

2.  Reduce  to  whole  or  mixed 
numbers  V- 

V-=-2|.  Ans. 

3.  Reduce  to  whole  or  mixed 
numbers  ^^. 

V-6J.  Ans. 


4.  Reduce  to  whole  or  mixed 
numbers  -^^. 

i^^9^.  Ans. 

5.  Reduce  to  whole  or  mixed 
numbers  -^j'j*. 

W=13i  Ans. 

6.  Reduce  to  whole  or  mixed 
numbers  ^^. 


TEACHERS 

EDITION. 

133 

7.  Reduce  to  whole  or  mixed 

11.  Reduce  to  whole  or 

mixed 

numbers  -y/-. 

numbers  ^^V/- 

W  =  I'iff  ■  -i^s- 

-Wi-  =  mi  ^ns. 

12.  Reduce  to  whole  or 

mixed 

8.  Reduce  to  whole  or  mixed 

numbers  -2//-. 

numbers  -\y-. 

%  =  13.  Ans. 

4M-  =  37.  Ans. 

13.  Reduce  to  whole  or 
numbers  -\^^. 

mixed 

9.  Reduce  to  whole  or  mixed 

-¥r  =  182V  ^^s- 

numbers  -^f f-^. 

14.  Reduce  to  whole  or 

mixed 

^fP  =  50ff.  Ans. 

numbers  ^^^f-. 

3^i-  =  18|0-.  Ans. 

10.  Reduce  to  whole  or  mixed 

15.  Reduce  to  whole  or 

mixed 

numbers  -yVy*- 

numbers  -^||-^. 

-V.¥  =  2G^v^^«. 

8  9|5=359.  ^ns. 

Exercise  XIX 

1.  Reduce   to   improper    frac 
3ns  3|. 

3|  =  V-  -^ns. 


2.  Reduce   to   improper   frac- 
tions 5j^^. 

3.  Reduce   to   improper    frac- 
tions 12y\. 

12t*x  =  Vt--  ^«"'- 

4.  Reduce   to   impro];)er    frac- 
tions lOy'^^. 


5.  Reduce   to   improper    frac- 
tions 8f. 

^^-^^.  Ans. 


6.  Reduce   to    improper   frac- 
tions 12^|. 

1211  =  -W-.  ^^^«- 

7.  Reduce   to    improper    frac- 
tions 84  J-^. 

8.  Reduce   to    improper    frac- 
tions 8tU-^f. 

864^f  = -s-^g^fi.  Ans. 


134 


ARITHMETIC. 


9.  Reduce   to   improper   frac- 
tions 41y§8^. 

10.  Reduce  to  improper  frac- 
tions 41t^^7^. 


11.  Reduce  to  improper  frac- 
tions 400H^. 

400H*  =  l^'^-.  Am. 


12.  E-educe  to  improper  frac- 
tions 50005§3.^. 

5000^^5  =  i^fger^.  Arts. 

13.  Reduce  to  improper  frac- 
tions lOOOOjf 

10000}!  =  J^f^.  ■^^^«- 

14.  Reduce  to  improper  frac- 
tions 300l7%3^. 


15.  Reduce  to  improper  fractions  73||. 

16.  Express  8,  7,  3,  5,  12,  13,  18,  29,  25  in  the  form  of  fractions, 
each  having  5  for  a  denominator. 

8  7  3  5  12  13  18  20  25 

=  ^.    =¥•    =-^^-    =¥•    =¥•    =¥•    =-?-•    =^^-    =^P. 

17.  Express  21  in  the  form  of  fractions,  having  for  denominators 
3,  5,  7,  8,  12,  13,  20.  25,  30,  37. 


21 

21 

21 

21 

21 

-¥. 

=  i§i. 

=  ^fl. 

=  ifA. 

=w- 

21 

21 

21 

21 

21 

-w. 

=w. 

=w. 

=  W- 

=w. 

18.   Express  12,  15,  23  in  the  form  of  fractions,  each  having  for 
denominators  12,  15,  23,  respectively. 

12  12  12  15  15 

-w.        -w.        -W-        =W-        -w- 

15  23  23  23 

-w.  -w.  -w.  =W- 


TEACHERS     EDITION. 


135 


Exercise  XX. 


1.  Reduce  to  lowest  terms  iff, 

1  2  0  _   1  5  _  5        Anc 

2.  Reduce  to  lowest  terms  |f f. 

3.  Reduce  to  lowest  terms  xWcJ- 

928     _llfi        yljio 
T32(7  —  T65-    ^'^S- 

4.  Reduce  to  lowest  terms  ||f  f . 

1728_216_108_12        /l^iQ 


6.  Reduce  to  lowest  terms  |f  l^. 

23]0_231_21_3        A^<. 
3  0  80  —  3  0  8  —  'IJ  —  ¥•    -^^S- 

7.  Reduce  to  lowest  terms  |^f ff. 

8.  Reduce  to  1  owest  terms  ^2%%% 


9.  Reduce  to  lowest  terms  -j|^f . 


10.  Reduce  to  lowest  terms  -^^^j. 

924    _231_77_11       A-nt 
10^2  —  273  —  ¥T  -  T3-     -^^S- 

11.  Reduce  to  lowest  terms  |f  |^. 
lift  =  f  f  =  f  •  ^ns. 

12.  Reduce  to  lowest  terms  x^A- 


13    Reduce  to  lowest  terms  f  fo  I- 

6732  _   1  683  _   153   _   1  7        Ay.^ 


14.  Reduce  to  lowest  terms  -oV/ifo  • 

6840     _171_19_1        J^a 
■Z7360  —  ■5'84  —  76  —  ¥•    •^^^• 

15.  Reduce  to  lowest  terms  f^f§. 

5760  _  576  _  144       /I.,, 
7170  0  —  70  0  —  TYZ-    ■^^''^■ 

16.  Reduce  to  lowest  terms  y^^f-jf. 

17.  Reduce  to  lowest  terms  ff"-!- 

18.  Reduce  to  lowest  terms  x^li- 

1  01  5  _   35       /l^o 

19.  Reduce  to  lowest  terms  2X^7- 

516     _   1  2        /J^a 
^T07  —  ?9-    ^^S. 

20.  Reduce  to  lowest  terms  -g^aVoV 

3  JL7  2     _  _3  5  2     _     3  2 
72807  —   8¥37  —  767- 

21.  Reduce  to  lowest  terms  ||m. 

78473)94653(1 
78473 
10 


16180 


1618 


809)78473(97 
7281 
5663 
5663 

G.  C.  M.  =  809. 


136 


ARITHMETIC. 


22.  Reduce  to  lowest  terms  ^ff  If. 

4|17o06 

4399)26145(5 


21995 


4150 


415 

83)17596(212 
1J86 
99 
83_ 
166 
166 
G.C.M.  =  83. 

23.  Re<]  vice  to  lowest  terms  |f  ^ff. 
44323)61087(1 


44323 

4 

1^764 

3 

4191 

1 

1397 

127)44323(349 
381 

622 
508 
1143 
1143 

G.C.M.  =  127. 


24.  Reduce  to  lowest  terms  i'jW. 

3 1 339 

113)1243(11 
1243 
G. CM. -113. 


25.  Reduce  to  lowest  terms  ^if?- 

1111177 

107)2675(25 
214 
535 
G.  CM.  =  107.        ^ 

26.  Reduce  to  lowest  terms  ji^H. 


3815 
.763 


G.  CM.  =  109)5123(47 
436 

763 
763 

27.  Reduce  to  lowest  terms  i-^  J  f  ^. 

14141)16289(1 
14141 
12|2148 

179)14141(79 
1253 


1611 
G.  CM.  =  179.  1^ 

28.  Reduce tolowesttermsff^^lf. 

29.  Reduce  to  lowest  terms  uWjTffV 
Divide  both  terms  Ijy  1001. 

30.  Reducetolowestterms^lf^^ 

Divide  both  terms  by  142857. 


TEACHERS     EDITION. 


137 


Exercise  XXI, 


1.  Find  the  product  of  f  x  2. 

§X^  =  -=U.  Ans. 
2 

2.  Find  the  product  of  f  X  9. 

3.  Find  the  product  of  10  X  |. 

2 

^X-  =  4.  Ans. 
1   ^ 


4.  Find  the  product  of  15  x  |. 

1^  X  ^  =  15.  Ans. 
1       3 

5.  Find  the  product  of  ^\  X  7. 

3 

-^X-  =  3.  ^ns. 
^;      1 

3 

6.  Find  the  product  of  16  X  |. 

2 

2^  X  -  =  10.  Ans. 
1      ^ 

7.  Find  the  product  of  f  X  2. 

I^l^l^li.  Ans. 
4 


8.  Find  the  product  of  ^^  X  5. 
|.  Ans. 


^X^ 

;^   1 

3 


Find  the  product  of  27  X  f . 

3 

^  X  -  =  15.  .4ns. 
1       9 


10.  Find  the  product  of  |f  x  2. 

^P      1      10        ^^ 
10 

11.  Find  the  product  of  ^§  X  3. 

20      1      20        '' 

12.  Find  the  product  of  |f  X  4. 

L3x^  =  l^  =  2|.  4ns. 
20      1       5        ^ 


13.  Find  the  product  of  5  x  \^. 

^xl^  =  ^  =  3i4ns. 
1      ^0      4        ^ 
4 


14.  Find  the  product  of  6  X  |f . 

Ll^  =  ^  =  3^.  4ns. 
1  ^^      10       ^*^ 
10 


138 


ARITHMETIC. 


15.  Find  the  product  of  7  X  H- 

1      20     20       ^^ 

16.  Find  the  product  of  8  x  f|. 

5 

17.  Find  the  product  of  1^  X  10. 


Exercise 


18.  Find  the  product  of  ^  X  12. 

gxLf  =  :,.». 

5 

19.  Find  the  product  of  ^  X  15. 

3 

13      I^     39     o'.     A 

4 

20.  Find  the  product  of  i^  x  20. 

I^xf  =  13.^n3. 

XXII. 


1.  Simplify  f  of  ^. 


2.  Simplify  ^  of  2^^. 

3 
^  X  2  1   -  ^  y  ^^  -  ^     Ans 


3.  Simplify  f  of  f . 


?X^  =  A    ^n, 
7     ^     21 
3 


4.  Simplify  2f  X  2^. 

6 
2|  X  2^  =  ^  X  I  =  G.  Ans. 


5.  Simplify  4|  X  2f 


4fx2|  =  |x^^ 


^  =  lOf  Am. 


6.  Simplify  4|  x  9f 


14 


-^-451.  .In*. 


7.  Simplify  ^  off  of  10. 


lx?xf.2.^n. 


teachers'  edition.  139 

8.  Simplify  |  of  f  of  f . 

-  X  -  X  -  =  -.  Ans. 

9.  Simplify  fxfXf  of  4i. 

2  3 

^X^X^xa-^X^X^X^-^-^-r     Ans 
5X-X-X4i--X-X-X--^-l^.  Ans. 

% 

10.  Simplify  \  of  ^\. 

2 

11.  Simplify  f  of  ^^  of  f  of  f  of  |  of  15f . 

%  9 

^x  ^  x^x^x  1x153 -^x  ^  x^x^x^x^^-^'^-l^     An, 

5 

12.  Simplify  5f  X  8f . 

2 

13.  Simplify  |  X  f  X  jV  X  7^ 

-X-X-X7^--X-X-X----l3.  ^r^s. 

14.  Simplify  f  of  |^  of  ^%  of  8f 

5 
^x^^x  ^  x8i-^x^^x  ^  x^^-^    ^ns 
3        2 

15.  Simplify  x«T  X  M  X  If  X  2ij 


11      21      48        ''      11      ^/      ^^      19     627       ^^^ 
3 


140  ARITHMETIC. 


16.  Simplify  If  XtV^xI^^it-  9 

43X  — Xl^^^--X^^^X^^^-^^^.  ^m. 

n    m 
22 

17.  Simplify  f  X  iM  X  H  X  17. 

^    ;?;    ^^    ;     11     ^^ 

11    ;7 

18.  Simplify  If  X^fX  MX  Ifi 

;^    4     3 

38x^X^Xl|i  =  ^X^X^X^  =  i  =  li  ^n. 
39     57     86       '^     3^     ^7     ^^     )J3     3       ^ 

;3    3     ;z 

19.  Simplify  i  of  |  of  |  of  f  of  f  of  f  of  |  of  |  of  1%  of  10. 

Iv^X^X^X^X^X^X^X^X^-l    Ans 

20.  Simplify  ^  of  ^^  of  30. 

5 

21.  Simplify  Hf  X  ^  X  H  X  If. 

3 

il^x-^^vl^xU-^^^X-^X^X^-^    Ans 

71      ;z      5     2 


22.   Simplify  |  x  f  X  /r  X  f  of  |  of  |  of  8. 
^x^xl-x^x^x^x^-'*^ 


23.  Simplify  tV  of  Ji;  of /jiV- 

3     ;3 

5     3 


teachers'  edition.  141 

24.  Simplify  j\  X  y'a  X  f  f  X  48. 

xx^'x^^^^  X 

25.  Simplify  \l  of  i^  of  ||  of  12. 

3 

?2x^X^X^-^    ^ns 

^0><^^^^^^l-4^"'- 

^    ^    ;^ 

4  ? 

26.  Simplify  If  of4|of  f. 


27.  Simplify  2fx  If  XllfX  8. 

2       4 
2tXlfXll|X8  =  |x|x|xf  =  l|^  =  52A.  ^n. 

3 

28.  Simplify  3f  of  2\  of  l^^^  of  l^V 

?  3 

3fx2^Xl^Xlx\  =  f  x|x|xl|  =  ^  =  20A.  ^ns 


29.  Simplify  H  X  5^  X  4^  X  A  X  5. 

Iix5,x4,x|x5  =  f|x|xixlxf  =  f=S2,... 

30.  Simplify  f  of  ^  X  8f  X  ^^  of  li|. 

2       2       7 
^x-^x82x-^xP7_^x  ^  x^^y  ^x^^--"^    Ans 
5  9 


142 


ARITHMETIC. 


31.   Simplify  HxHxHf. 

9        i% 

^%     38     m     7ti 

2  n 

17 


— .  Ans. 


32.  Simplify  Iff  Xf If  X^V^. 

4     xn     8 
m  m  im  405' 

l^^       9  9 

5 

33.  Simplify  \m  of  ,%  of  i§|f  • 

2 

;zT^^    ;2;^^    im    243' 
9        9       ;;2 

3 


Exercise  XXIII. 


1.  Divide  ff  by  6. 

24,6  =  1x^^4  ^n. 
35  ^     35     35 

2.  Divide  |?  by  5. 

3.  Divide  f  by  8. 

3     o      1      3      3      . 

7  8      7     56 


4.  Divide  18f  by  7. 

8 

184  +  7  =  1  X  ^  =  22    An^ 
^  T      3        ' 

5.  Divide  \  by  f . 

5     3     ^^5     n     . 
--^-  =  -x-  =  -.  Am, 
8     4     3     ^     ti 


6.   Divide  \\  by  f . 

16     8     ^     n 
i 


7.  Divide  1}  by  3f 
ia      oi      7      10 


Q  7        Ol 


teachers'  edition.  143 


8. 

Divide  5^  by  4f . 

7 

9. 

Divide  8f  by  4^. 

2 

«»-^  =  f-f  =  f.x¥--- 

10. 

Divide  71  by  4f. 

6 

^       ^5        7      30      5      25        '^ 

11.  Divide  6f  by  9^. 

63.91  =  27^19^1^27^27^^^^ 
^        '       4       2       19      ^      38 

2 

12.  Divide  8f  by  4|. 

13 

8.2-^41  =  ^^^  =  1x^  =  ^  =  15-    Ans 

'      '     3-3     ;^^  3     7      '• 


^ns. 


13.  Divide  3|  by  if 

^      '^      9      27     ;^  ^       2 
2 

14.  Divide  4f  by  6f. 

43.6f  =  ^.^2^1x^  =  ^.  ^n. 
^       ^7       9      j?^      7      14 
2 

15.  Divide  5  by  4f . 

6 

16.  Divide  3f  of  2i  by  li  of  2^. 


34of2i-liof2^  =  l^of?^|ofl^  =  ^X?X?X^  =  3.  Ans. 


2     2      9      ^     ;i     3     ;^ 


144  ARITHMETIC. 


17.  Divide  2f  by  3^  of  1^. 

2 

18.  Divide  2^^  of  5^  by  7f . 

19.  Divide  5f  of  8^  of  1^  by  2^^  of  5|. 
.|of8^oflf^2^ofr>|  =  |of|of^^?lof| 

=  1x^x11x^^x1  =  ^^  =  6^.  ^n.. 
^  "^  ^  "^  J      ^^     ^0      7 
7       ? 

Exercise  XXIV. 

1.  Express  with  least  common  denominator  ^,  ^,  ^. 
L.C.D.  =  2x3x5  =  30. 
1     2    5      15     12    25 


2    5    6  30 


Ans. 


2.  Express  with  least  common  denominator  |,  f ,  |,  -j^^. 
L.C.D.  =  23x3'»x5  =  360. 
2    5     7     9      240    200    315    324 


3'    9'    8'    10  360 


Ans. 


8.  Express  with  least  common  denominator  ^,  ^,  ^^^i-,  ^f. 
L.C.D.  =  2'x3x5x7  =  840. 
5    1      5     19     700    105    200    V^a 


6'    8'    21'    35  840 


Ans. 


4.  Express  with  least  common  denominator  •^,  ^'^,  ,^,  ^. 
L.  C.  D.  =  22  X  3>  X  5»  =  800. 
2      7      3      8      120    315    108     160 


15'    20*    25'   45  800 


Am. 


teachers'  edition.  145 

5.  Express  with  least  common  denominator  |f,  ij,  -^f,  }f. 
L.  C.  D.  =  2=5  X  3  X  52  =  600. 
12     17     13     19      288     255     130     152 


25'    40'    60'    75  600 


Ans. 


6.  Express  with  least  common  denominator  |,  -^^,  /^,  ^^,  |f. 
L.  C.  D.  =  23  X  3  X  5  X  7  =  840. 
3      7      4      3      19     315     196     96    90    665 


8'    30'    35'    28'    24  840 


-.  Ans. 


7.  Express  with  least  common  denominator  |-^,  -^j,  |f .  ff ,  ||. 
L.  C.  D.  =  2*  X  33  X  5  =  2160. 
11      7      13     23     17      1485     840     1404     1()56     680 


16'    18'    20'    30'    54  2160 

8.  Which  is  the  greater,  if  or  ^|  ?   |  or  |  ?   |  or  j^^  ? 


Ans. 


L.C.D.  =  22x52 

=  100. 

L.C.D.-2x32  = 

=  18. 

L.C.D.  =  5xl2  =  6a 

13      65 
20     100' 

5_15 
6      18' 

3_36 
5     60' 

17      68 
28      100' 

7_14 
9      18' 

7  _35 
12     60' 

.-.  Il"  is  the  greater. 

.-.  f  is  the  greater. 

.•.  1  is  the  greater. 

9.  Arrange  the  fractions  j\,  j-^,  ^f  in  order  of  magnitude. 
L.  C.  D.  =  2^  X  3'^  =  72. 

L  =  ^        11  =  ^         1^  =  ^.  1^     1    11    Ans 

12     72'        18      72'        24      72*  24'    12'    18' 

10.  Arrange  the  fractions  j\,  j%,  x*r,  j\  in  order  of  magnitude. 
L.  C.  D.  =  22  X  32  X  5  X  11  =  1980. 
A    A    ±     i  _H25     1056     720     770 
12'    15'    11'    18  1980 

A,    1,    A,    A.  Ans. 
11     18     12     15 


146 


ARITHMETIC. 


Exercise  XXV. 


1.  Find  the  sum  of  J  +  f . 

1  +  3^^  =  2.  ^n. 

2  2     2 

2.  Find  the  sum  of  i  +  f  +  i 

3  3     3     3       ^ 

3.  Find  the  sum  of  ^  +  ^  +  f . 

l+i  +  §  =  5  =  U.  Ans. 

4  4     4     4^ 

4.  Find  the  sum  of  1^  +  2^. 
H  +  2i  =  3lJJ^  =  4.  ^ns. 


5.  Find  the  sum  of  1|  +  2f . 
H+2f-  =  3iJi  =  4.  ^m. 

6.  Find  the  sum  of  3^  +  f . 
3^  +  1  =  31^  =  4.  Ans. 

7.  Find  the  sum  of  2|  +  3|. 
2!  +  3|  =  5^  =  6f  ^ns. 

8.  Find  the  sum  of  1|  +  f . 


9.  Find  the  sum  of  ^j  +  ^j  +  {^  +  H- 

17     17     17     17      17       ^^ 

10.   Find  the  sum  of  8^^  +  6^j  +  5\^  +  }f 

8i^T  +  6^7  +  5|^  +  1^  =  19f|  =  21^7.  Ans. 


11.   Find  the  sura  of  |  +  f 
5     6  30  ^ 


12.  Find  the  sum  of  |  +  |. 

3  +  7^6  +  7^1j.^n. 

4  8         8  ^ 


13.   Find  the  sum  of  i  +  i- 
3  +  1      2 


1  +  1 

2  G        6 


Ans. 


14.    Find  the  sum  of  ^  +  ^. 
L.  C.  D.  =  2''  X  3  X  5  =  60. 

^^^''-'l.Ans. 


60 


60 


15.    Find  the  sum  of  /j  +  ^\. 
L.C.D.=.2*X3  =  48. 
15  +  22     37 


48 


48 


-.  Ans. 


16.   Find  the  sum  of  12»  +  7^. 
121 +  7A- 191^4 -19H.^»w 


17.   Find  the  sum  of  85,5j  X  27|J. 

85/j  X  27H  -  n2H1P  -  113^-  ^^' 


TEACHERS     EDITION. 


147 


18.  Find  the  sura  of  ^  +  ^  4-  ^  +  i. 

L.  C.  D.  =  22  X  3  X  5  =  GO. 

30  +  20  +  15  +  12  _  rz  _  -,  1 7 
~60~    ''■ 


60 

19.  Find  the  sum  of  i  +  |  +  f  +  |. 

L.C.D.  =  2'^X  3x5  =  60. 
30  +  40  +  45  +  48      163 


Ayis. 


60 


60 


m  ^ns. 


20.  Find  the  sum  of  |  +  j^  +  j%  +  ^\  +  ^l 

L.  C.  D.  =  22  X  3  X  5  =  60. 

50  +  55  +  32  +  21  +  26  _  184  ^  3  1 
60  60        ^^ 

21.  Find  the  sum  of  5^}  +  HM 

L.  C.  D.  =  2 


Ans. 


^  +  3%  +  17tV  + 14  +  n^v 


¥0' 


22.  Find  the  sum  of 

94  ^  151^  +  163^1  +  111  ^  ify^ 

L.  C.  D.  =  22  X  32  X  7  =  252. 
1984|f  =  199||.  Ans. 

23.  Find  the  sum  of 

31  +  41  +  11  +  2. 
L.C.D.  =  2  +  3  +  5  =  30. 

24.  Find  the  sum  of 

h\  +  2-A  +  5/^  +  j%. 
L.  G.  D.  =  22  X  3  X  52  =  300. 

25.  Find  the  sum  of 

f  +  lf  +  2  +  3f  +  4^3j. 

L.  C.  D.  =  23  X  33  X  7  =  504. 

10HI  =  llfM-  ^n^- 


h24 

X  3  X  52  =  600. 

85/o%.  Ans. 

26.  Find  the  sum  of 

4i  +  3f  +  2f-  +  li  +  ,?j. 

L.  C.  D.  =  23  X  32  X  7  =  504. 

10fM=lli§i.  ^m. 

27.  Find  the  sum  of 

H  +  ^V  +  lO  +  ff. 

L.C.D.  =  23x3x5x7  =  840. 

10||f.  Ans. 

28.  Find  the  sum  of 


29 


L.  C.  D.  =  2*  X  32  X  52  =  3600. 

UU 

=  nU-  Ans. 

.   Find  the  sum  of 

2  +  1  + 

If  +  4|  +  5i|. 

L.C.D.= 

=  23x32=72. 

12W  = 

=  14f i.  Ans. 

148 


ARITHMETIC. 


30.  Find  the  sum  of 

L.C.D.  =  2*x  5x11  =  880. 
IG-VsV—IStI^.  ^^«- 

31.  Find  the  sum  of 

L.  C.  D.  =  23  X  32  X  5  =  360. 
CIM  =  81|.  Arts. 

32.  Find  the  sum  of 

^j  +  j'i  +  n- 

L.C.D.  =  2x7x11  =  154. 

33.  Find  the  sum  of 

20A  +  ll^V  +  5i  +  305. 
L.  C.  D.  =  23  X  3  X  5  =  120. 

34.  Find  the  sum  of 

H  +  i^  +  H- 

L.  C.  D.  =  2--'  X  3  X  19  =  228. 

35.  Find  the  sum  of 

iV  +  H  +  M  +  if- 

L.C.D.  =  2-^x3xl7  =  204. 


36.  Find  the  sum  of 

3l7|  +  17^r  +  4r%  +  /3  +  6|  +  ^. 

L.C.D.  =  2x3x5xl7  =  510. 

344-W^  =  346|H.  Am. 

37.  Find  the  sum  of 

4A  +  82\  +  4i\  +  5f  +  5|  +  f. 
L.C.D.  =  3x5x7x11  =  1155. 

26f!-H  =  2%\V  ^^• 

38.  Find  the  sum  of 

H  +  ^^  +  ^^h  +  n  +  hih- 

L.  C.  D.  =  2880. 
17tm  =  18HH-  ^^«- 

39.  Find  the  sum  of 

4A +  73\  +  5H  + 275^3/, +  2§f 

L.C.D.  =  2-^x3x7xl3  =  1092. 

293Hfl  =  294^1.  ^^• 

40.  Find  the  sum  of 

H  +  Vi  +  6^  +  400^  +  51H. 

L.C.D.  =  23x7x3x11  =  1848. 

464J3|i  =  465^H-  ^^• 


Exercise  XXVI. 


1.  Find  the  vahio  of  52J 
52^-46  =  0^.  Ans. 


46. 


2.  Find  the  value  of  ^  -  f 
6     3     6-3     3      1 


9     9 


9     3 


Am. 


3.  Find  the  value  of  f  -  f 
3_2_9^ 
4 


-8-1.  Am. 
12        12 


4.  Find  the  value  of  ^^  —  •^. 
A  _  A     32  -  25     2. 
15      12"      60      "60' 


.^714. 


TEACHERS     EDITION. 


149 


5.  Find  the  value  of  {^  —  j\. 
11  3  ^77-27^  50  __  25 
18      14         126         126      63 

Ans. 

6.  Find  the  value  of  4  —  |. 

4-i  =  3i.  Ans. 

7.  Find  the  value  of  7  -  f . 

7_|  =  6f  Ans. 

8.  Find  the  value  of  3  —  |. 


9.  Find  the  product  of  8  —  f . 


10.  Find  the  product  of  5  —  f . 

5-4  =  4^.  Ans. 

11.  Find  the  value  of  5  -  |. 

5  —  I  =  4f .  Ans. 

12.  Find  the  value  of  6^  -  5^. 
61-5^  =  1-2-^  =  1^  Ans. 


13.  Find  the  value  of  4f  -  3f-. 
4|-3f=ll-V5-  =  M-  ^^«- 

14.  Find  the  value  of  7^  -  2tV. 

^  1        9  3—  f:  1  0-9  _  K  1        A„^ 

15.  Find  the  value  of  7f  —  4|. 
7|  _  4|  =  3J-\^0-  =  2||.  Ans. 

16.  Find  the  value  of  6|  -  2|. 
6f  -  2|  =  4-^-/ =  3H-  ^ns. 

17.  Find  the  value  of  9f  -  4f . 
94  _  4|  =  524^-^25  ^  429..  ^^3. 


18.   Find  the  value  of  4f  -  ^. 


-^^  -  4^.  Ans. 


19.  Find  the  value  of  6|  -  4f . 

6|-4|  =  2-V2'-  =  2xV 

20.  Find  the  value  of  7J  -  2|. 
71  _  2J  =  5-^f^  =  4|.  Ans. 

21.  Find  the  value  of  8^  -  4f 
8^-4t  =  4^F  =  3H. 


22.   Find  the  value  of  85-2^  -  27H- 

85,V  -  27H  =  58-^V~  =  57111  =  57|§.  Ans. 

24.   Find  the  value  of  10  -  3|. 
10 -31- =  61.  Ans. 


23.   Find  the  value  of  8yV  -  m- 
8tV-2H  =  6^-V(P  =  6^V^^s- 


25.  Find  the  value  of  120||  -  llOif . 

120fi  -  llOif  =  10^^//^  =  101^.  Ans. 

26.  Find  the  value  of  5i|  -  11 


150  ARITHMETIC. 


28.  Find  the  value  of  2^f  J  -  If «3. 

9151  _  1  103  _  16.0.4- 8J_5  _  749       >!„, 

29.  Find  the  value  of  4  -  l^Ul 

4  -  imi  =  2^VKru¥^ = -mi  ^^«- 

30.  Find  the  valuo  of  1473  -  279H. 

1473- 279 j-|=1193tV  Ans. 

31.  Find  the  value  of  1473^5^  _  279j^. 

1473/j  -  279i^  =  ii94^^^lg^  =-- 1193^.  Ans. 

32.  Find  the  value  of  1473^V  -  279^- 

1473^^5  -  279H  =  1194J-\-/^  =  1193^|.  Ans. 

33.  Find  the  value  of  278  j|  -  30^^. 

278H  -  30-i5j  =  248^^f^  =  248||.  Ans. 

34.  Find  the  value  of  125/j  -  10^. 

125A  =  lOU  =  115^^^  =  1 14tf  Ans. 

35.  Find  the  value  of  118/r  -  17^^. 

36.  Find  the  value  of  94^^  -  91|f. 

9'h«^r-91||  =  3i^VV^  =  2,Vr-  Ans. 

37.  Find  the  value  of  7^r  -  2\l 

7^  -  2H  =  5^^  =  m  Ans. 

38.  Find  the  value  of  ^  -  |f 

235  _  13     235  -  91      48      . 
357     51 "      357     "  119* 


39. 

Find  the  value  of  ^|- 

-j\\- 

17 

29 

204  -  203 

63 

108 

756 

40. 

Find  the  value  of  /^  - 

-AV 

9 

38 

43 
209 

99-86 
418 

teachers'  edition.  151 


= .  Ans. 

756 


13  A 
=   .  Ans. 

418 


41.  Find  the  value  of  iff  -  fff . 

146  268  __  1032  -  804  _  218 
273  637     1911     1911' 


Ans. 


359  199  _  1795- 1791  _  1   ^^^^ 

360  200     1800     450'   ^'^* 


Exercise  XXVII. 

1.  Simplify  3f  -  2|  +  4^^^  +  1|  -  5^^. 

Sum  of  plus   terms  =  9f  § 
Sum  of  minus  terms  =  S-^i^ 

Difference  =  l|i;.  Ans. 

2.  Simplify  lA  -  H  +  n-  -  2i  -  m 

Sum  of  plus   terms  =  8|| 
Sum  of  minus  terms  =  4}^ 

Difference  =  3if-^-.  Ans 

3.  Simplify  12  -  3f  -  1^3^  -  4^V  +  2M  -  4|. 

Sum  of  plus   terms  =  14|^ 
Sum  of  minus  terms  ==  IS^Mjc 


Difference  =    l^.Ans. 

4.   Simplify  433-V  -  H  -  Ifi  -  1||  -  2^1  -  2/^  -  2f f  -  3,^. 
Sum  of  plus   terms  =  43y'3- 
Sum  of  minus  terms  ==  16/5- 

Difference  =  27f ^.  Ans. 


152  ARITHMETIC. 


5.  Simplify  ^  +  ^  +  7/^  +  8^1  +  7i  +  8^^^  +  4^^  -  36^^- 

Sum  of  plus   terms  =  36^^ 
Sum  of  minus  terms  =  865^1^ 

Difference  =    0.  Ans. 

6.  Simplify  (S^j,  +  H^  +  17^  +  40)  -  (30^  +  1 1 H)- 

Sum  of  plus   terms  =  66j§f 
Sum  of  minus  terms  =  41| 

Difference  =  2,5r^^g.  Ans. 

7.  Simplify  (172}f  +  QSyi^^)  +  172||  -  93^^^- 

(17211 +  93TV7)  +  (172f|-933^,\) 
=  172||  +  93tVt  +  172^1  -  93tVt 
=  172||  +  1 72|f  =  3441^^^  =  344^^  A,us. 

8.  Simplify  (172|f  +  93x^A)  -  (172f|  -  93  AV)- 

(17211 +  93i»A)-(172|f-93TV\) 
=  172|f  +  93^T-V- 17211 +  93^ 
=  93^tV  +  93^  =  186-iVt.  ^^■ 


9.   Simplify  (t3j_^)  +  (^j  +  ^|^). 


U3     39/      \,78      156/ 


13     39     78      156 
36-8  +  10  +  7     15     . 
=  156 =  52-^"^- 

10.  Simplify  ^  -  ^  _  2|  +  3f  +  7/j  -  If  -  3^. 

Sum  of  plus   terms  =  lljf 
Sum  of  minus  terms  =    4y  ^ 

Difference  =   6ff|.    Ans. 

11.  Simplify  T^^-y^^-T^^-T-^^^^. 

_3^_  J 9___5 3000  -  700  -  90  -  5  ^  441 

10     100     1000     10000"  10000  "200U 

12.  Simplify  9|- 7- f-f. 

9J-7-i-^-2li4|-iil=.l2V.  Ans. 


.ins. 


teachers'  edition.  153 

13.   Simplify  5|  +  8f  -  If  -  4|. 

Sum  of  plus   terms  =-  14j\ 

Difference  =  8j|^.  Ans. 


Sum  of  minus  terms  =    6f  ^ 


14.   Simplify  6|  -  5f  +  4|  -  4/j. 

Sum  of  plus   terms  =  llg^ 
Sum  of  minus  terms  =  lOj^ 

Difference  =    1^-^.  Atis. 

15.  Simplify  14tV  +  9|  -  6|  -  12|  -  3f . 
Sum  of  plus    terms  =  23|| 

Ans. 

2f-9f  +  10x%-14TV 
Sum  of  plus   terms  =  30|f 
Sum  of  minus  terras  =  26|f 

Difference  =  43^%.  Ajis. 

17.  Simplify  95f  -  9/^  -  8f  -  14^%  +  74f . 
Sum  of  plus   terms  =  169| 
Sum  of  minus  terms  =    32 A 


Difference  =  137.^||.  Ans. 

18.  Simplify  12|  +  23|  -  {4j%  +  12f  +  Tif). 
Sum  of  plus 
Sum  of  minus  terms 


Difference  =  llx^^jj.  Ans. 

+  ^^\  +  I'iA). 
Terms  outside  pareij thesis  =  MjW^ 
Terms  inside   parenthesis  =  28|5-§ 

Difference  =    5/'X.  Ans. 


154 


ARITHMETIC. 


20.  Simplify  97f  -  (20  +  9|  +  18^  +  24f^). 
Terms  outeide  parenthesis  =  97 J 
Terms  inside  parenthesis  =  72t'\^ 

Difference  =  25^\y.  Am. 


21.  Simplify  2}i  +  3Jf  -  {l^  +  IH  +  II), 

Terms  outside  parenthesis  =  6^ 
Terms  inside  parenthesis  =  ij^^^n 

Difference  =  l^^iVj.  Ans. 

22.  Simplify  ill  +  flH  -  m^- 


143     2471 
100     1000 


82643       143000  +  247100  -  82»i43 


100000 


100000 


307457 
100000 


Ans, 


Exercise  XXVIII, 


1. 

Simplify 

^ 

Multiply  by  44 

100 
165 

33 

2. 

Simplify 

3 

Multiply  by  8. 

24 

57 

-l.An.. 

3. 

Simplify 

Hi. 

131 

Multiply  by  21. 

360. 

-If.  Am. 

4.  Simplify  -L 


Multiply  by  27. 


225 


1.  ^n. 


5,  Simplify  —^. 


Multiply  by  99. 
253      11      . 

4r4-r8-^'"- 


6.  Simplify  liil?i. 

Hof^ 

2  2 

r7'';<3'^y    2l 


TEACHERS     EDITION. 


155 


7.  Simplify  1^. 

Multiply  by  72. 
180-112^68^^        ^^^^ 
132-117     15        '^ 


8.  Simplify  l^iiiy. 

Multiply  by  280. 
2912-480^2432^^82 
1995-861      1134       ^^^ 


9.  Simplify  iA^. 


10.  Simplify  ?i:ii4. 
^    ^    21  +  13 

Multiply  by  84 

567-114     453 


182  +  120     302 


1|-.  Ans. 


1.  Simplify  M. 

9 

8^_;2^^     39_351_ 

=  lOJW 

If       35      ^^      35 

-4ns. 


12.  Simplify  ^ 


4      ;^     3     ^     8     12     24* 


Ans. 


13.  Simplify  iiof?i. 


=  -.  Ans. 

7 


14.. Simplify  5i 


m 


m 


^=lAns. 
175     5 


15.  Simplify  ?i±?l. 

^=^lJl^^.Ans. 

2 


16.  Simplify  3^^. 


n^  9      n^  7_189_22« 
2  ^m. 


17.  Simplify  ^^  +  T7  +  A  +  7, 

Multiply  by  60. 
51  +  44  +  42  +  48 


51-44  +  42 


41  —  91 
18.  Simplify  ^^^—^5 


185.  Ans. 


Multiply  by  28. 
116  -  63       53 


182-60      122 


Ans. 


156 


ARITHMETIC. 


19.  Simplify  ?l^^>— i- 

Multiply  by  280. 


749  -  1280  +  875 


344     8     . 
=  -.  Aru. 


1640-1365  +  112     387     9 


20.  Simplify    U  X  1»  +  j  of  2^- H  X  2. 


H  +  f-4!_45-f-21-26 
if  +  f-H     26  +  21-45 


^  =  20.  ^n,. 
2 


21.S.n.pUfy2ix^i^ix,JA_- 


9     516 

4     297  +  368 


236      38 
"^140 


H 

7     •  5 


22.  Simplify  Jj  -  7f  +  5f -4|  ^ 

7455  -  6600  +  4900  -  4032      1723 
8316  -  7448  +  6615  -  5760  ^  1723 


1,  Aiu. 


Exercise  XXIX. 


1.  What  fraction  of  8  is  3? 
3 


;.  An». 


2.   What  fraction  of  3  is  8  ? 
|-2f  Am. 


3.   What  fraction  of  9  is  7? 
7 
9' 


I.  Ant, 


4.  What  fraction  of  7  is  9? 
7 


5.   What  fraction  of  8  is  12? 
12 


=  H.  Ana. 


6.   What  fraction  of  12  is  8  ? 

8      2     . 
—  =■-.  Am. 
12     3 


7.  What  fraction  of  2^  is  |. 

a   What  fraction  of  |  is  2^? 
^  -  3f .  Afu, 


TEACHERS     EDITION. 


157 


9.   What  fraction  of  2|  is  1^? 
2|      11 

10.   What  fraction  of  1^  is  2f  ? 
21 


U 


2|.  ^ns. 


11.   What  fraction  of  2^  is  7f 
2i        ^^ 


13.   What  fraction  of  7^  is  2|  ? 

—5^  = .  Ans. 

71      176 

13.    What  fraction  of  31  is  8|? 


14.    What  fraction  of  $  2  is  $  1^  ? 
3 
$2      4 


m  =  -.   Ans. 


15.  What  fraction  of  $  2^  is  $  5  ? 

1^=2.  Ans. 

16.  What  fraction  of  $f  is  fp 

$1      3 

17.  What  fraction  of  $f  is  $|? 

$1     10 


18.  What  fraction  of  |2f  is  ||? 

$2|      33 

19.  What  fraction  of  $^  is  l^^^  ? 

I^  =  --  ^ns. 
$i      5 


20.   What  fraction  of  $  1  is  $  |  ? 

11  =  7 
$1      8' 


=  X  Ans. 


21.  What  fraction  of  $  10  is  |  f  ? 

1 10      15 

22.  What  fraction  of$100  is  $6? 

$  100     50 

23.  What  fraction  of  $100  is  $4^? 

ML  =^.  Ans. 
$100     200 

24.  What  fraction  of  $4  is  $25  ? 

$25 
$4 

95.   What  fraction  of  100|  is  8f  ? 

^==^.Ans. 
lOOf      905 

26.   What  fraction  of 
21isif  of3|? 

mii^lAns. 
21  7 


158 


ABITHMETIC. 


27.  What  fraction  of 

18iHi8|of33|? 

28.  What  fraction  of 

3^i8fxH? 


H 


15 


29.  Wliat  fraction  of 

3^x52>7i8l720? 


1720 


3iVx5A 


llOHf  Am. 


30 

What  fraction  of 

3ixfof^i8lf? 

^^             ^      im 

3^  X  f  X  t      10 

31. 

What  part  of 

||x|fi8ix4xf? 

^-■^- 

32.   What  part  of 
ISfXlX/^isIoflflofli? 

ixillxii.l.  ^,, 
131  X|X^^ 


33.   What 


part  oi  U+H+^js  +  i  is  U-H  +  tV  -  I? 


Multipl 
-44  +  42-48^_l_,  ^,, 
+  44  +  42  +  48      185 


Multiply  by  60. 
51-44  +  42-48_  1 
51      '•      '- 


34.  What  part  of  4|  -  2^  is  6^  -  2^  ? 

6i_:^^182-60^12_2_^^       ^^ 
4^-2i      116-63      53        ^' 

35.  What  part  of  17f  -  12f  is  5  -  ^  -  ,^  -  2^  ? 

171 -12f      • 

Multiply  by  6825. 

34125  -  525  -  700-  273     32627 


120575-87750 


32825 


Ans. 


86.   What  part  of  24  -  17,^  is  7  +  ^^  -  ^  -  Ji? 

.24-17A    * 
Multiply  by  5266. 
36856  +  702  -  325  -  1 287     35945     ,  , . ,      . 


1263ti0- 91126 


35235 


TEACHERS     EDITION. 


159 


fx2TV     ;/      3      3      3^      9 


38.   What  part  of 

("-ihYiih-")- 

V4^~63|)-^  (737  +  41:1) 
_  (14  -  ft)  ^  {^P-  -  13)  _  -W  X  A 


(If -1)^(11  +  1) 


226  V  45 
TTl   -^  53 


i^f.  ^ns. 


Exercise  XXX. 


1.  Keduce  to  common  fractions 
in  their  lowest  terms  0.125. 

0.125  =  tV¥o  =  h  ^^s- 

2.  Reduce  to  common  fractions 
in  their  lowest  terms  0.625. 

0.625  =  T«j,V^  =  f.  Ans. 

3.  Reduce  to  common  fractions 
in  their  lowest  terms  0.675. 


0.675  =  tW, 


Ans. 


4.  Reduce  to  common  fractions 
in  their  lowest  terms  10.864. 

10.864  =  10xVo%  =  lOitf  •  ^^s- 

5.  Reduce  to  common  fractions 
in  their  lowest  terms  50.84. 

50.84  =  50xV%  =  50|i.  Ans. 


6.  Reduce  to  common  fractions 
in  their  lowest  terms  3.00025. 
3.00025  =  3^ 

7.  Reduce  to  common  fractions 
in  their  lowest  terms  8.1075. 

8.1075  =  8^1^^^^  =84§^.  Ans. 

8.  Reduce  to  common  fractions 
in  their  lowest  terms  35.01024. 


35.01024 


35T^§M^  =  35^ff^. 
Ans. 


9.    Reduce  to  common  fractions 
in  their  lowest  terms  7.015625. 


10.   Reduce  to  common  fractions 
in  their  lowest  terms  20.100256. 

20.100256 = 20j^^o_o^%%=  20/xWi7. 
Ans. 


160  ARITHMETIC. 


11.  Reduce  to  common  fractions  in  their  lowest  terms  10.012575. 

10.012575  =  10t^^^^^  =  10j^M7-  ^^«- 

12.  Reduce  to  common  fractions  in  their  lowest  terms  104.236. 

104.235  =  104.,^  =  1042^.  Am. 

13.  Reduce  to  common  fractions  in  their  lowest  terms  50.0004. 

50.0004  =  50tt^  =  50^^.  Am. 

14.  Reduce  to  common  fractions  in  their  lowest  terms  lOO.OOl. 

100.001  =  lOOy^.  Am. 

16.   Reduce  to  common  fractions  in  their  lowest  terms  8.00725. 
8.00725  =  SjMj^j,  =  8jH7r-  ^ns. 

16.  Reduce  to  common  fractions  in  their  lowest  terms  20.018375. 

20.018375  =  20jmih  =  ^0^^.  Am. 

17.  Reduce  to  common  fractions  in  their  loweet  terms  125.6048. 

125.6048  =  125T^^^=125fH.  Am. 

18.  Reduce  to  common  fractions  in  their  lowest  terms  0.128. 

0.128  =  tV^  =  ^.  Am. 

19.  Reduce  to  common  fractions  in  their  lowest  terms  0.73125. 

0.73125  =  tWi^  =  IH-  ^^• 

20.  Reduce  to  common  fractions  in  their  lowest  terms  1.1876. 

1.1875 -l^Wj^^lt*,.  Am. 

21.  Reduce  to  common  fractions  in  their  lowest  terms  0.603125. 

0.603125 -VWWW- iff  ^^• 

22.  Reduce  to  common  fractions  in  their  lowest  terms  6.03125. 

6.03125 -6yHf^-6iV.  ^ns. 

23.  Reduce  to  common  fractions  in  their  lowest  terms  60.3126. 

60.3125 -60AVW-60A.  Am. 

24.  Reduce  to  common  fractions  in  their  lowest  terms  7.0316. 

7.0315 -7x*i^-7,Sb.  Am, 


TEACHERS     EDITION. 


161 


Exercise  XXXL 


1.  Reduce  to  decimals  |. 

0.875 
8)7.000 

2.  Reduce  to  decimals  \^. 

0.9375 


16)15.0000 


3.    Reduce  to  decimals  -j^^, 
0.28125 


32)9.00000 


4.   Reduce  to  decimals  ^\. 
0.36 
25)9.00 


5.   Reduce  to  decimals  ^\. 
0.078125 


64)5.000000 

6.  Reduce  to  decimals  4^V(7- 

0.01375 
8)0.11000 
4.01375.  Ans. 

7.  Reduce  to  decimals  ^-^^ioi^. 

0.00015625 


32)0-00500000 
5.00015625.  Ans. 

8.   Reduce  to  decimals  9^^|f  (j. 
0.0048046875 
256)1.2300000000 
9.0048046875.  Ans. 


9.   Reduce  to  decimals  ll^f^. 
0.00475 


4)0.01900 
11.00475.  Ans. 

10.  Reduce  to  decimals  yf^. 

0.072 
125)9.000 

11.  Reduce  to  decimals  ^^^^. 

0.00425 


4)0.01700 

12.  Reduce  to  decimals  ^^f. 

0.9296875 
128)119.0000000 

13.  Reduce  to  decimals  -^\^\. 

0.0208 


625)13.0000 


14.    Reduce  to  decimals  ^^j. 
0.04296875 


256)11.00000000 


15.   Reduce  to  decimals  yf^. 
0.01875 


16)0.30000 


16.   Reduce  to  decimals  Jj^. 
7.75 
16)124.00 


162 

AEITHMETIC. 

17.  Reduce  to  decimals  |  of  If. 

19.   Reduce  to  decimals  3f  of  4f 

3 

2x2  =  ^. 
^^5     5 

1.2 
5)6.0 

2 

X?^37_74 
d""  9       5' 
14.8 
5)74.0 

18.   Reduce  to  decimals 

foffof^. 

4^8^X0     64 

2 

20.   Reduce  to  decimals  f  f  of  f  f . 
29     49      1421 
32     64     2048' 

0.328125 

0.69384765625 

64)21.000000 

2048)1421.00000000000 

Exercise  XXXII. 

1.  In  like  manner  simplify  7^  +  4f  +  9^  +  llff . 
7^  +  4|  +  9^^  +  11|| ,  7.4  +  4.625  +  9.65  +  11.90625  =  33.58125.   (1) 

n+n+m-^  hh = ^mi = 33^ = 33.58125.  (2) 


2.  In  like  manner  simplify  84^  +  19^  -  f  J. 

84^  +  19H  +  f*  =  84.65  +  19.523809^|  +  0.82  =  104.993809^.      (1) 
84H  +  19ii  +  fi -  I03l8ggtim-fl722  =.  10311^  =  104fW 

-  104.993809ii.  (2) 

3.  In  like  manner  simplify  4»|  +  13^  +  42f^  +  2^  +  li 

4rj  +  13  J^  +  42|^  +  m  +  li  =  4.421875  +  13.85  +  42.74  +  2.8125  + 1.5 

-  65.324375.  (1) 
m  +  mi  +  42H  +  2i|  +  H -  6287g-M8g0^}ifitl80Qt800 

-  ^^m  -  ^W^  -  65.324375.         (2) 

4.  In  like  manner  simplify  5J  +  13f  +  19^  +  7^. 

5J  +  13f  +  19^  +  7^"  5.876  +  13.8  +  19.4375  +  7.15  -  46.2626.    (1) 

6H  13f  +  19^  +  7A  =  44lJi±Ai^-*iP±ia  -  44W 

-  46|^  -  46.2626.  (2) 


teachers'  edition.  163 

5.  In  like  manner  simplify  5^^  +  f  of  If  +  |  of  2^  +  f  of  f . 

5T'(T  +  fXl|  +  |x2f  +  fxf 

=  5.5  +  0.666f  X  1.8  +  0.875  x  2.285714f  +  0.75  x  0.625 

=  5.5  +  1 .2  +  2  H-  0.46875  =  9.16875.  (1) 

5A  +  |Xlf  +  |x2f  +  -|xf:=5i  +  li  +  2  +  M-8''igr^' 

=  811^  =  ^7  =  9.16875.  (2) 

6.  In  like  manner  simplify  1^^  of  2f . 

lyi  X  2|  =  1.4166f  X  2.625  =  3.71875.  (1) 

7 
1AX2|  =  1^X|  =  ^^  =  311  =  3.71875.  (2) 

4 

7.  In  like  manner  simplify  Sy^  +  2|-f , 

h%  +  2M  =  3.3125  +  2.95  =  6.2625.  (1) 

h%  +  2M  =  5^W-  =  5-W-  =  6f  ^  =  6.2625.  (2) 

8.  In  like  manner  simplify  7f  —  4f . 

7f  -  4f  =  7.4  -  4.625  =  2.775.  (1) 

7f-4f  =  3J-VTp  =  2fi  =  2.775.  (2) 

9.  In  like  manner  simplify  82^  —  37||^. 

82^  -  37H  =  82.2  -  37.6875  =  44.5125.  (1) 

82^  -  37H  =  45J-%-/-5-  =  44|^  =  44.5125.  (2) 


m- 


10.  In  like  manner  simplify  100  —  17|i 

100  -  17iif  =  100  -  17.1808  =  82.8192.  (1) 

100  -  17HI  =  82f  if  =  82.8192.  (2) 

11.  In  like  manner  simplify  5^  —  1|  of  1||. 

5^  -  1^  X  HI  =  5.5  -  1 .5  X  1.5416f  =  5.5  -  2.3125  =  3.1875.  (1) 

5^  -  1|  X  HI  =  5^  -  2j\  =  3^  =  3.1875.  (2) 


164 


ARITHMETIC. 


12.  In  like  manner  simplify  i|  —  ^\. 
^  _  ^1  _  0.56  -  0.171875  =  0.388125. 

H  -  H  -  "iVoV^  =-  im  -  0.388125. 


(1) 
(2) 


13.   In  like  manner  simplify  8^  —  1 J  x  t^. 

8^  _  IJ  X  T^  =  8.2  -  1.5  X  0.1875  =  8.2  -  0.28125  =  7.91875.  (1) 
8i  -  H  X  A  -  Si  -  /j  =  8^^  =  7||*  =  7.91875.  (2) 


14.    In  like  manner  simplify  ^|  X  1000. 
j^f  X  1000  =  0.29H875  x  1000  =  296.875. 
j^j  X  -i^V^  -  ^«^  =  296f  =  296.875. 


(1) 
(2) 


Exercise 

1.  Reduce  to  decimals  f . 

0.5 

2.  Reduce  to  decimals  ^. 

0.45 
lll5!00 


3.  Reduce  to  decimals  3^. 

0.416 
12)5.000 
3t»jj- 3.416.  Am. 

4.  Reduce  to  decimals  ^. 

0.183 
6)1.100 

6.  Reduce  to  decimals  3^. 

0.35416 
48)17.00000 
3H  -  3.35416.  Afu. 


XXXIII. 
6.   Reduce  to  decimals  2/y. 

o.iss 


37)5.000 
2/7.  =  2.1 35.  Ans. 

7.  Reduce  to  decimals  y^^. 

0.00081 
37)0.03000 

8.  Reduce  to  decimals  ll^J. 

0.13095238 


84)11.00000000 
11^=1113095238.  Ans. 

9.   Reduce  to  decimals  9^^^. 

0.10185 
108)11.00000 
9iVr  -  9.10185.  Ant. 


TEACHERS     EDITION. 


165 


10.  Reduce  to  decimals  11 35. 

0.1 142857 
35)4.0000000 
1X3%  =  11.1142857.  Ans. 

11.  Reduce  to  decimals  |^f . 

0.267857142 
56)15.000000000 

12.  Reduce  to  decimals  ^*y. 

0.380952 


21)8.000000 

13.   Reduce  to  decimals  |f. 
0.39 


33)13.00 


14.   Reduce  to  decimals  f-J. 
0.5285714 
7)3.7000000 


15.  Reduce  to  decimals  2^-^^. 

0.22745098039215686 
255)58 .00000000000000000 
2^5/^  =  2.22745098039215686. 

16.  Reduce  to  decimals  5^\. 

0.230769 


13)3.000000 
5^6^  =  5_3_  ^  5.230769.  Ans. 


17.    If 


117 


be  expressed  as  a  decimal,  the  quotient  will  contain 
how  many  decimal  places? 

As  7  is  the  highest  power  of  2  or  5  in  the  denominator,  and  as 
there  are  no  otl^r  factors  than  2  or  5,  there  will  be  seven  decimal 
places  in  the  quotient. 


18.   If 


119 


be  expressed  as  a  decimal,  how  many  decimal  places 


25  X  13 
will  precede  the  recurring  period  ? 

As  5  is  the  highest  power  of  2  or  5  in  the  denominator,  and  as 
there  is  another  factor  than  2  or  5,  five  decimal  places  will  precede 
the  repetend. 


19.  If 


57 


5=^x7 


be  reduced  to  a  decimal,  how  many  decimal   places 


will  precede  the  recurring  period  ? 

As  2  is  the  highest  power  of  2  or  5  in  the  denominator,  and  as 
there  is  another  factor  than  2  or  5,  two  decimal  places  will  precede 
the  repetend. 


166 


ARITHMETIC. 


Exercise  XXXIV. 


1.   Reduce  to  common  fractions 
in  their  lowest  terms  0.245. 

0.245  =  ^-tV^.  Ans. 


2.  Reduce  to  common  fractions 
in  their  lowest  terms  0.425. 

3.  Reduce  to  common  fractions 
in  their  lowest  terms  53.00243. 


53.00243 


53^^  =•  53yT%^. 
Ans. 


4.   Reduce  to  common  fractions 
in  their  lowest  terms  7.2011. 

7.26li  =  ?§§&§.  Am. 

6.   Reduce  to  common  fractions 
in  their  lowest  terms  2.5306. 

2.5306 -2iM*-2iHi  ^^• 


6.   Reduce  to  common  fractions 
in  their  lowest  terms  0.00426. 

0.00426  -  ^mn  =  lUh-  ^'M. 


7.   Reduce  to  common  fractions 
in  their  lowest  terms  31.203. 

31.203  =  31^^  =  31^.  Am. 


8.   Reduce  to  common  fractions 
iu  their  lowest  terms  0.35i. 

0.35i  =  fH  =  if  Am. 


9.   Reduce  to  common  fractions 
in  their  lowest  terms  1.416. 

1.416  =  lH^  =  ltV.  Am. 


10.    Reduce  to  common   frac- 
tions in  their  lowest  terms  0.5575. 

0.5575  -  1^^  =  \%\.  Am. 


11.   Reduce  to  common  fractions  in  their  lowest  terms  2.081. 
2.08  U  2^ -2^.  Am. 

18.  Reduce  to  common  fractions  in  their  lowest  terms  5.12297. 
5.12297  =  5iHM  =  5/iV  ^^«- 

18.   Reduce  to  common  fractions  in  their  lowest  terms  0.3590 
0.3590- If  e-T^.  Am. 

14.   Reduce  to  common  fractions  in  their  lowest  terms  4.3162. 
4.3162 -4HH-4iH-  ^'W. 


teachers'  edition.  167 


15.  Reduce  to  common  fractions  in  their  lowest  terms  0.7283. 

0.7283  =  |f|f  =  f Iff.  Ans. 

16.  Reduce  to  common  fractions  in  their  lowest  terms  5.142857. 

5.142857  =  5ifffM  =  55VW^V    Ans. 

17.  Reduce  to  common  fractions  in  their  lowest  terms  0.2368. 

0.2368  =  ff If  =  Hff-  Ans. 

18.  Reduce  to  common  fractions  in  their  lowest  terms  1.136. 

1.136  =  lift  =:1tV8  =  1A-  Ans. 

19.  Reduce  to  common  fractions  in  their  lowest  terms  1.53i. 

1.53i  =  lf|i  =  l,-5,-V  Ans. 

20.  Reduce  to  common  fractions  in  their  lowest  terms  3.28963. 

3.28963  =  3f  If  If  =3/2\V     Ans. 

21.  Reduce  to  common  fractions  in  their  lowest  terms  5.8783.  - 

5.8783  =  5Un-^U-  Ans. 

22.  Reduce  to  common  fractions  in  their  lowest  terms  1.69408. 

1.69408  =  If f Iff  =  Hif If  Ans. 

23.  Reduce  to  common  fractions  in  their  lowest  terms  0.48324. 

0.48324  =  fff^f  =  ff|.  Ans. 

24.  Reduce  to  common  fractions  in  their  lowest  terms  0.00i2213. 

O.OOi2213  =  ^iffif^  =  TT¥tVVo-  Ans. 


168 


ARITHMETIC. 


Exercise  XXXV. 


1.  Find  the  G.C.M.  and  L.C.M. 

H-i 
G.C.M.  of  7,  14,2        -1. 
L.C.M.  of  9.  27,  5        -105. 
.-.  G.  C.  M.  of  fractions  =  y^^. 
L.  C.  M.  of  7,  14,  2        =  14. 
G.C.M.  of  9,  27,  5        ^^  1. 
.*.  L.  C.  M.  of  fractions  =  14. 

2.  Find  the  G.C.M.  and  L.C.M. 
of2i2i:^. 

2f-^,2f-V,:^  =  ^. 
G.  C.  M.  of  20,  12,  1  =1. 
L.  C.  M.  of  9,  5,  10  =  90. 
.'.  G.  C.  M.  of  fractions  =  ^. 
L.  C.  M.  of  20,  12,  1  =  60. 
G.C.M.  of  9,5,  10  =1. 
.-.  L.  C.  M.  of  fractions  =  60. 


3.  Find  the  G.C.M.  and  L.C.M 
of  33f ,  50f 

33^  =  ifi,  oOf  =  ^^, 
G.C.M.  of  234.  405       =9. 
L.C.M.  of     7,      8       =56. 
.-.  G.  C.  M.  of  fractions  =  ^g. 
L.C.M.  of  234,  405      =10,530. 
G.C.M.  of     7,      8       =1. 
.-.  L.  C.  M.  of  fractions  -  10,530. 

4.  Find  the  G.C.M.  and  L.C.M. 

G.  C.  M.  of   7,  35,  49     =  7. 
L.C.M.  of  24,  36,  60     =-360. 
.•.  G.  C.  M.  of  fractions  =  y||y. 
L.  C.  M.  of   7,  35,  49     =  245. 
G.C.M.  of  24,  36,  60     =12. 
.-.  L.  C.  M.  of  fractions  =  Y/ 


5.  Find  the  G.  C.  M.  and  L.  C.  M.  of  5^,  7^,  8^,  4f ,  9^  Q^j. 

5i,  7i  8 J,  4f ,  9i  6A  -  Jjji,  V.  ^,  V.  ¥.  H- 
G.C.  M.  of  11,  22,  33,  44,  55,  77  =  11. 
L.C.M.  of  2,  3,  4,  9,  6,  12  =33. 

.-.  G.  C.  M.  of  fractions  =  ^i. 

L.  C.  M.  of  11,  22,  33,  44,  55,  77  =  4620. 
G.  C.  M.  of  2,  3,  4,  9,  6, 12  -1. 

.-.  L.  C.  M.  of  fractions  -  4620. 

6.  Find  the  G.  C.  M.  and  L.  C.  M.  of  f  f  ^,  |,  ^  ^,  ^. 

G.C.M.  of  1,1,  1.1,  1,1,  1  -1. 

L.  C.  M.  of  2,  3,  4,  5,  6,  10, 12  -  60. 

.'.  G.  C.  M.  of  fractions  —  ^. 

L.C.M.  ofl,  1.1,  1,1,  1,1  =1. 

G.C.M.  of2,  3.  4.  6,  6,  10,  12  -1. 

.-.  L.  C.  M.  of  fractions  -  1. 


teachers'  edition.  169 

7.  Find  the  G.  C.  M.  and  L.  C.  M.  of  50^,  67^  44|,  841  707. 

50|,  67i  44f ,  84|,  707  =  ifi,  ^-p,  ^^,  ^^,  707. 

G.  C.  M.  of  101,  202,  404,  505,  707  =  101. 

L.C.M.  of     2,      3,      9,      6,      1  =  18. 

.'.  G.  C.  M.  of  fractions  =  V/  =  Hi- 

L.  C.  M.  of  101,  202,  404,  505,  707  =  14,  140. 

G.C.M.  of     2,      3,  .9.      6,      1=1. 

.-.  L.  C.  M.  of  fractions  =  14,  140. 

8.  Find  the  G.  C.  M.  and  L.  C.  M.  of  |,  |,  f ,  |,  f ,  j%. 

G.  C.  M.  of  4,  5,  6,  7,8,    9  =  1. 
L.  C.  M.  of  5,  6,  7,  8,  9,  10  =  2520. 
.•.  G.  C.  M.  of  fractions       =  -^^-^^jj. 
L.  C.  M.  of  4,  5,  6,  7,  8,  9    =  2520. 
G.C.M.  of  5,  6,  7,  8,  9,  10=1. 
.'.  L.  C.  M.  of  fractions       =  2520. 

9.  Find  the  G.  C.  M.  of  1^\,  l^f ,  ^,  f  f 

lTV.H!.4f.|f  =  iiM,-^^,|f. 

G.  C.  M.  of  15,  40,  30,  25  =  5. 

L.C.M.  of  14,  21,    7,42=42. 

.-.  G.  C.  M.  of  fractions     =  ^%. 

L.  C.  M.  of  15,  40,  30,  25  =  600. 

G.  C.  M.  of  14,  21,    7,  42  =  7. 

.-.  L.  C.  M.  of  fractions     =  -6fo  =  85f. 


10. 

Find  the  G.  C.  M.  and  L.  C.  M.  of  18f ,  57| 

18|  =  -%\  57^  = 

^i^- 

G.  C.  M.  of  92,  115 

=  23. 

L.C.M.  of   5,      2 

=  10. 

.-.  G.  C.  M.  of  fractions 

=  H- 

^\%. 

L.C.M.  of  92,  115 

=  460. 

G.  C.  M.  of    5,      2 

=  1. 

.-.  L.C.M.  of  fractions 

=  460. 

170  ARITHMETIC. 


11.  Find  the  G.  C.  M.  and  L.  C.  M.  of  134f ,  128^,  115^. 

134f,  128^,  115^  =  ^.  ^^,  -4^. 

G.C.M.  of  539,  385,  231  =  77. 
L.C.M.  of4,  3,  2  =12. 

.-.  G.  C.  M.  of  fractions     =  H  =  ^i- 
L.  C.  M.  of  539,  385,  231  =  8085. 
G.C.M.  of  4,  3,  2,  =1. 

.-.  L.  C.  M.  of  fractions    =  8085. 

12.  Ftnd  the  G.  C.  M.  and  L.  C.  M.  of  2^,  If  |,  ^^V 

G.C.M.  of  72.  112,  63  =1. 

L.  C.  M.  of  25,  75,  100  =  300. 

.'.  G.  C.  M.  of  fractions  =  y^^. 

L.C.M.  of  72,  112,  63  =1008. 

G.C.M.  of  25,  75, 100  =25. 

.♦.  L.  C.  M.  of  fractions  =  J^  =  40^. 

13.  A,  B,  and  C  start  together  and  travel  round  a  circular  island, 
in  the  same  direction.  It  takes  A  2\  days,  B  2f ,  C  2|  days  to  walk 
round  the  island.  They  travel  until  they  all  meet  at  the  point  of 
starting.  In  how  many  days  will  they  be  together  at  the  point  of 
starting? 

2i2|,2J  =  l.-V,^. 

L.C.M.  of  7,  17,  23  =  2737. 
G.  C.  M.  of  3,  6,  8     =  1. 
/.L.C.M.  =2737. 

2737  days.  Am. 

14.  If  the  step  of  a  man  be  2^ft.,  and  that  of  a  horse  be  2fft.,  find 
tlio  smallest  number  of  feet  which  is  an  exact  number  of  man-paces 
and  of  horse-paces. 

L.C.M.  of  7,  11  =  77. 

G.C.M.  of  3,  4    -1. 

.-.L.C.M.  -77, 

77  ft.  Ans. 


teachers'  edition.  171 

15.   Find  the  largest  number  that  is  contained  without  remainder 
in  2f ,  6tV,  Hi  and  19^ 

2f ,  6tV,  Hi.  I9i  =  ¥.  W.  ¥.  H^' 

G. CM.  of  23,  115,23,  115  =  23. 
L.C.M.  of9,  18,  2,  6  =18. 

.-.  G.C.M.  =ff  =  lTV 


Exercise  XXXVI. 

1.  Simplify  iui  nuh  zm-h,  im- 

2709  _^         43785  _  973  2436   ^    203  4087  ^  67^ 

6966      is'         56835      1263*         567216     47268*         5063     83* 

2.  Which  is  greater,  and  how  much,  |  or  |f  ? 

7    19     56,     57  19  .  ,     u      1 

-,    —  =  — ! -.  .-.   —  IS  greater  by  — 

9    24  72  24      ^  -^  72 

3.  Find  the  sum  of  3|,  2t*t,  5^  7tV,  l^V- 

H  +  2A  +  5H  7^^  + 1^\  =  18^+^0^,5+7  7+15  ^  182  3  1  =  20tV.  Ans. 

4.  Simplify  5i-3f  +  2T%- If. 

Sum  of  plus    terms  =  8f . 
Sum  of  minus  terms  =  5-^^. 

Difference  =  3^|. 

5.  Simplify  If  +  3f  -  2^^  +  4.^%  -  3^^. 

Sum  of  plus    terms  =  9f^. 
Sum  of  minus  terms  =  6^^^. 

Difference  =  3^-^. 


6.   Simplify  ii+ii. 


3H3t^42  +  46^88^^_,     ^ 
4i-2^^     52-31      21      ^2T-  ^^«. 


172  '  ARITHMETIC. 


7.   Simplify   the   expressions:      7^2|;    ^-    ^;    15-j-f;    ^^ 

7A^9;    43i^37i;    f^  ;    5|  ^  4f  ;    l-^l^  ;    106 -.- 8f  ;    i]-- 
10^  t  X  f  ^7 

?5i_  11x151  =  2101  ^11^^       5f.4t  =  Axf  =  |  =  H; 


2     3     ,5     45     „„.  t2fli  =  ?x|x^x2=9 


S^Jj-     95       2        11^ 

^^■*-5  =  I^T  =  f  ==22i;  |xf      ?";2-    7'   3     14 

1ft       ^       Iff 


7jL*9-lx«-9  il-l2xH  =  52?  =  3S* 

8.  Simplify  the  expressions ;  7J|  X  8 ;  43Ji  x  6| ;  6^  -.-  8^ ;  5^  X  51 

HofM;  HofAofJoffoa;  HoffW;  ix|xAx4xf 

7Hx8  =  ||xf  =  f  =  60t. 

4 
573 
43Hx6|  =  2|2x|-5p  =  286i; 

■«**«i-^xf  =  -^  '   ^4 

;y  ^z   11* 

33 

11 


lx?x^x?x?  =  l. 
3 


3 

6lV 

X51 

'> 

1 

■258; 

}|^ 

,11^ 
^13" 

_121. 
"l56' 

52 

r 

i 

xf- 

'k- 

teachers'  edition.  173 


9.  By  what  must  |-  be  multiplied  to  obtain  |  ?  |  to  obtain  f  ?  ^-  to 
obtain  |  ?  f  to  obtain  |  ?  |  to  obtain  |  ? 

2  ■  6     1     ^        '  3 

3  '  6     13        '  2 

7.  3_5     7_35_.,i 
8  •  5~3^8~24~^^- 

10.  By  what  must  ^  be  divided  to  obtain  |?  |  to  obtain  ^?  f  to 
obtain  f  ?  |  to  obtain  f  ?  f  to  obtain  |?  8  to  obtain  7^1? 

6"21^3'  8"548     32       ^" 

2  37_8^3     24 

2^1  =  ^X^  =  4.  5"^8~7''5-35  = 

3     6     13' 

6^6"^''^-^'  ^'^^^^-243''l~243-^^^^- 

11.  What  number  exceeds  5f  by  4|  ? 

5|  +  4|  =  91-^2^-^  =  9|f  =  10/3. 


12.  From  what  must  6f  be  subtracted  to  leave  |  of  3^  ? 

14 

2''^^-^^¥-¥-^^' 

6|  +  lf  =  7^%2_5  =  7||  =  8^V 

13.  What  fraction  falls  short  of  ^V  by  -i^  ? 

7       3  _  35  -  9  ^  26  ^  13 
12     20         60         60     30' 

14.  What  fraction  is  that  to  which  -^^^  must  be  added  to  give  ^  ? 

11       5  ^44-15^  29 
57     76        228         228' 


174 

ARITHMETIC. 

15.   Convert  into  decimals 

i;  i;  i;  f:  ii 

ii;  iV.A.A. 

i^.^.H.H.H; 

if;  f:  1; 

A:  ^• 

2)1.0 

8)7.000 

0.0625 

6)1.00 

0.5 

0.875 

16)1.0000 
0.0625 

9 

0.16 

4)1.00 
0.25 

0.5625 

6)5.00 

3)1.0 

0.0625 

0.0625 
11 

0.83 

0.3 

3 

0.6875 

7)3.000000 

4)3.00 

0.1875 

0.42857i 

0.75 

0.0625 

0.0625 

9)5.0 

8)1.000 

5 

13 

0.5 

0  12.^ 

0.8125 

V.  1  L^J 

8)3.000 

0.3125 

11)3.00 

0.375 

0.0625 

0.0625 

0.27 

8)5.000 

7 

15 

4)0.700 

0.626 

0.4375 

0.9376 

0.175 

16.   Convert  into  common  fractions:   0.16;   0.016;   0.125;   0.13; 

0.725;  0.625;  0.00625;  0.8125 

;  0.03125  ;  0.08  ;  0.54 

;  0.016 ;  0.5437 ; 

0.027;  0.277;  0.68494;    1.345. 

0.16 

=  iW 

=  A; 

0.03125  =.Vytfy^ 

=  ^ 

0.016 

=  T*fT7 

=  Th; 

0.08       =Tk 

=  3^ 

0.126 

-iVW 

=  i; 

0.54       =M 

=  A 

0.13 

=  tV^: 

0.016     =^ 

=  7^ 

0.725 

-i^ 

-H; 

0.5437   ^\m 
0.027     =  ^V 

=  TfTr; 

0.625 

-im 

-*; 

0.277     =  \%% 

=  A; 

0.00626 

-TlW^ 

r  =  Tb; 

0.68494  =»H|^^ 

=  IMH; 

0.8126 

-iVrfW 

=  11; 

1..345     =  \\\l 

-iH- 

17.  Simplify  '^ 

X2.27 

1.136 


M><JiZ  .  2i2lM  _  "  X  Hf  X  I  -  f  -  6i  -  5.6.  4«. 
1.136  1>V         5      M     ??      6      ^ 


teachers'  edition.  175        ^ 


18.  Multiply  6.954  by  5.303,  and  express  the  result  as  a  whole 
number  and  common  fraction. 

51 
6.954  =  6fi;     1^^175^8925^33213 
5.303  =  5if;'     22       33       242  '^^" 

19.  Simplify  li  of  2f  +  6|  -f-  2f  and  reduce  the  result  to  a  decimal. 

Hx2fH-6|-^2|  =  fx-V*-  +  T\X-V-  =  4i  +  2^  =  6^5_6/^  =  6.7. 

20.  From  what  number  can  4^|  be  taken  9  times  and  leave  no 
remainder  ? 

4Hx9  =  :^xf  =  l|-l  =  40i 
4 

21.  Of  what  fraction  is  17|  the  7th  part  ? 

17ix7  =  ^x|  =  ^  =  12U. 

22.  Add  I  0.35,  f,  f,  0.il2,  45.28. 

f  +  0.35  +  I  +  f  +  0.112  +  45.28 

=  0.8  +  0.35  +  0.625  +  0.75  +  0.112  +  45.28  =  47.917. 

23.  Convert  into  decimals  H;  A;  J%<  H;  11;  A;  ^V 

0.86                 0.27  0.1 142857                   0.283 

15)13.                ll}3^  35)4.0000000            60)l7.000 

0.736842105263157894  0.384615          '     0.1320754716981 

19)  14.000000000000000000  13)5.000000         53)7.0000000000000 

24.  What  part  of  }f  is  j^\j  1 

1241  '  73     ;^     JLUl     85' 
5        17 

25.  Divide  0.0015  by  0.012,  and  express  the  result  as  a  common 
fraction  in  lowest  terms. 

0.125 


12)1.500  0.125  =  \. 


176  ARITHMETIC. 


26.  Convert  into  decimals :  s\  ;   jy^^  ;   j| ;   \. 

0.09375  0.00009375  0.2297  0.141857 

32)3.00000  32)0.00300000  74)17.0000  7)1.000000 

27.  If  the  product  of  two  factors  is  |,  and  one  factor  is  1^,  find 
the  other  factor, 

8       *     ^^     2 


28.  If  the  dividend  is  |J  and  the  quotient  6^,  find  the  divisor. 

12     ^    13    ;;z    78 

6 

29.  The  dividend  is  12||,  quotient  3,  remainder  l^^  ;   find  the 
divisor. 

(12H-lA)-^3  =  10M-H3  =  ixW  =  m  =  3Mi 

30.  Find  the  G.  C.  M.  and  the  L.  C.  M  of  833,  1127,  1421,  343. 

71833        1127        1421        343 

71119  161  203  49 

17  23  29  7 

G.  C.  M.  =  7  X  7  =  49. 
L.  C.  M.  -  73  X  17  X  23  X  29  =  3,889,277. 

31.  Arrange  in  order  of  magnitude  ^j,  f f ,  if,  ^,  f §. 

.'.  the  order  of  magnitude  is  ^,  ^,  |f  ^f ,  f^. 

32.  Find  the  L.  C.  M.  of  {^  |f ,  ^j. 

L.  C.  M.  of  15,  26,    65  =  390. 

G.C.M.  ofl7,  51,  102-17. 

.'.  L.C.M.  of  fractions  =  ^7^  =  22|f . 

33.  Find  the  G.  C.  M.  of  f  f ,  b^,  f  ^,  and  6^. 

G.C.M.  of  65,  39,  91,  13-13. 
L.C.M.  of  68,    2,64,    2  =.1088. 
.'.  G.  C.  M.  of  fractions    -  yif,. 


teachers'  edition.  177 

34.   Convert   into   common   fractions   in   lowest    terras:    7.2011; 
.954;    5.303;    21.396. 

7.201  i  =  7|»ff.  6.954  =  6|ff    =  6|i. 

5.303    =5f|f  =  5if.  21.396  =  21f  11=213*3-4.. 

3^ 


"'"''"'•'     51-71^28^  +  1 

nxlj\  +  ^,\-3j%     4  +  4tV-3A     4ff     7 

5i-7|-^28^V  +  i         5i-T\  +  i        51      8 

36 

Simplify  ^t^^^^X  2^ -7i 
3^  +  21-4^^ 

6f +  5ix3}-7i     6|  +  17f-7i 

^  +  ^-^T\         3^  +  2|-4tV 

_  945  +  2420  -  1015     2350      ^^  , ., 
448  +  350-574        224          ^'^' 

37. 

-»"^^ff?^i- 

2f-li  +  9T-V       616-330  +  2000      2286     2 

41-2^+13/j-      924-495  +  3000     3429     3 

38. 

Simplify  (3-71- 1-^08)  X7.03_ 
2.2 -3^% 

(3.71  -  1.908)  X  7  03      1.802x7.03      12.66806     ,  ,,,^0 

2.2-,^                  2|-^               2           "■""^""' 

39. 

Simplify        ^t^t       ^.^fHof4i 
^     ^   Hof-l-^lOi"^       13|of5i 

5f-^f       ^2.fHof4i 

l^off-5-10|      '       13|of5| 
=  i^x^x^x'^x^x^x^x^^x   ^   x^ -2^^-423 

^  '^2''^'^g'<  3  '^^''^'^y'^?;r;'^i6~"64  ~  ''^• 

2  3 


178  ARITHMETIC. 


40.  Simplify  H  of  2^  +  6|  i-  2|  +  (s^  +  ^:^^±^\ 

I .',  of  2U  6J  -H  2ii   f  ['^\   s  ^A2i±0:^\  =.  4 1  +  2^  +  5^  +  M 
V  2.2-0.G4/  If 

=  4i  +  2i  +  5|  +  Mi  =  ll^iii^i^^W^±^ 
=  llHH=12Hf 

41.  Simplify  0.9  of  f  of  f  of  15|. 

0.9o£Joffofl5i  =  ^^x2xfxf  =  fl  =  5A. 
2      2 

42.  What  part  of  f  18  i- 

12     2     4 

43.  What  part  ofO.390625  is  0.05? 

16 
0.05     ^_^_^^..  1_16 
0.390625     fl     25     ?0     125 
5 

44.  0.09  is  what  fraction  of  0.2045? 

0.09    _    -h   ^tV_4 
0.2045     \m     ^?     9' 

46.   Convert  into  decimals  |f  ;  |f  ;  f|. 

0.731343283582089552238805970149253 
67)49.000000000000000000000000000000000 
0.378  0.84931506 

37)14.000  73)62.00000000 

46.   Tlio  Tf .  0.  M.  of  three  numbers  is  15,  and  their  L.  C.  M.  is  450 
What  are  tho  numbers? 

450-(5x3)x2x3x5 
5X3-16-G.C.M. 
15x2-30. 
15  X  3  -  46. 
16x5-76. 


TEACHERS     EDITION. 


179 


EXEECISE 

XXXVII. 

1. 

Reduce  3  yds.  2  ft.  to  inches. 

5.   Reduce    82,976,432   in.  to 

3  yds.  2  ft. 

miles. 

X3 

12 

82976432  in. 

9 
2 

3 
5i 

6914702  ft.  8  in. 

2304900  yds.  2  ft. 

11  ft. 
Xl2 

132  in. 

2 

11 
320 

4609800                          [yds. 

419072  rds.  8  half-yds.  =  4 

1309  mi.  192  rds. 

2. 

Reduce  4  mi.  124  rds.  to  feet. 
4  mi.  124  rds. 

1309  mi.  192  rds.  4  yds.  2  ft.  8  in. 

X320 

6.   Reduce   7   mi.    3^  yds.   to 

1280 

inches. 

124 

7  mi.  3|  yds. 

1404  rds. 

X1760 

Xl6i 

12320 

23166  ft. 

H 

3. 

Reduce   27  rds.  4|yds.  to 

123231  yds. 

inches. 

X36 

27  rds.  ^  yds. 

X5^ 

443646  in. 

148i 
X4| 

7.   Reduce  27  mi.  222  rds.  to 

inches. 

152^  yds.    • 

27  mi.  222  rds. 

* 

X36 

X320 

5499  in. 

8640 

4. 

Reduce  290  leagues  to  feet. 

222 

290  leagues. 

8862  rds. 

X3 

X5i 

870  knots. 

48741  yds. 

X6086 

X36 

5294820  ft. 

1754676  in. 

180 


ARITUMETIC. 


8.  Eedace  712  mi.  to  inches. 

712  mi. 
X5280 


3759360  ft. 
Xl2 

45112320  in. 


9.   Reduce  540,451  ft.  to  miles. 
31540451ft. 
5i|i80150yd8.  1ft. 
2 


11 
320 


360300 
32754  rds.  6  half-yards 


3  yds. 


102  mi.  114  rds. 
102  mi.  114  rds.  3  yds.  1  ft. 

10.   Reduce  271,256  in.  to  miles. 


12 
3 

271256  in. 
22604  ft.    .  . 
7534  yds.    . 
2 

*.  .  8  in. 
.  .  2  ft. 

11 

15068 

320 

1369  rds..  . 

.  .  9  half-yards  = 

4  mi    . 

.  89  rds 

4  mi.  89  rds. 

4^  yds.     2  ft.  8  in 
i  yd.  =  1  ft.  6  in 

4|  yds 


4  mi.  89  rds.  5  yds.      1  ft.  2  in. 


11.   Reduce  723,964  ft.  to  miles. 
31723964  ft. 


5^1241321  yds.  1ft. 

2 
11 
320 


482642 


43876  rds.  6  half-yards  -  3  yds. 


137  mi.  36  rds. 
137  mi.  36  rds.  3  yds.  1  ft. 


TEACHERS     EDITION. 


181 


12.    Reduce  233,205  in.  to  miles. 


12 

233205  in. 

3 

19433  ft 

9  in. 

^ 

6477  yds.  .  .  . 
2 

2  ft. 

11 

12954 

320 

1177  rds.  .  .  . 

7  half-yards  =  3^  yds. 

3  mi.    ... 

217  rds. 

3  mi.  217  rds.  3^  yds.    2  ft.  9  in. 

^  yd.  -  1  ft.  6  in. 

3  mi.  217  rds.  4 

yds.    1  ft.  3  in. 

13.  Reduce  10  chains  to  inches. 

15.    If  the  height  of  a   horse 

10  ch. 

be  16  hands,  how  many  feet  is 

X4 

his  height  ? 

16  hands 

40  rds. 

X4 

Xl6^ 

12)_64  in. 

660  ft. 

5  ft.  4  in. 

Xl2 

7920  in. 

16.   If  a  train  move  40  ft.  in  a 

second,  what  is  its  rate  in  miles 

14.    Reduce     233,185     in.    to 

per   hour  ?      (One    hour  =  3600 

fathoms. 

seconds.) 

12)233185  in. 

3600 
40  ft. 

6)19432  ft.  1  in. 

5280)144000  ft. 

27ift§  mi. 

3238  fath.  4  ft. 

3238  fath.  4  ft.  1  in. 

=  27Ami. 

Exercise 

XXXVIII. 

1.   Reduce  92,638  sq.  yds.  to 

2.   Reduce  1,223,527  sq.  in.  to 

inches.             Q2638  sq.  yds. 

yards. 

^                       X9 

144)1223527  sq.  in. 

833742  sq.  ft. 

9)8496  sq.  ft.  103  sq.  in. 

Xl44 

944  sq.  yds. 

1200588- 

t8  sq.  in. 

944  sq.  yds.  103  sq.  in. 

182 


ARITHMETIC. 


3.  Reduce  721  sq.  mi.  to  rods, 

721  sq.  mi. 
_X_640 

461440  A. 
Xl60 


73830400  sq.  rds. 

4.  Reduce  34,729  sq.  yds.  to 
rods. 

30})34729  sq.  yds. 

4 

121)138916 

1148  sq.  rds.  8  quarter- 
sq.  yds.  =  2  sq.  yds. 

1148  Bq.  rds.  2  sq.  yds. 

6.  Reduce  1  A.  to  feet. 

1  A. 
160 

160  sq.  rds. 
X30^ 

4840  sq.  yds. 
X9 


43560  sq.  ft. 


5.  Reduce  3  A.  107  sq.  rds. 
27  sq.  yds.  7  sq.  ft.  23  sq.  in.  to 
inches. 

3  A. 
Xl60 

480 
107 

587  sq.  rds. 
X  S0\ 


17756f 
27 


177831  sq.  yds. 
X9 


1600531 

7_ 

1600601  sq.  ft. 
Xl44 


23048748 
23 


23048771  sq.  in. 


7.   Reduce  99,894,712  sq.  in.  to  acres. 

144)99894712  sq.  in. 

9)693713  sq.  ft.  40  sq.  in. 
30J)77079  sq.  yds.  2  sq.  ft. 

4 

121)308316 
160)2548  sq.  rds.  8  quarter-yds.  =  2  sq.  yds. 
15  A.  148  sq.  rds. 

15  A.  148  sq.  rds.  2  sq.  yds.  2  sq.  ft.  40  sq.  in. 


TEACHERS     EDITION. 


183 


8.  Reduce  15,376  sq.  yds.  to  acres. 

30^)15376  sq.  yds. 

4 

121)61504 

160)508  sq.  rds.  36  quarter-sq.  yds  =  9  sq.  yds. 
3  A.  28  sq.  rds. 

3  A.  28  sq.  rds,  9  sq.  yds. 

9.  Reduce  562,934  sq.  in.  to  rods. 

144)562934  sq.  in. 

9)3909  sq.  ft.  38  sq.  in. 
30^)434  sq.  yds.  3  sq.  ft. 

4 

121)1736 

14  sq.  rds.  42  quarter-sq.  yds.  =  10|^  sq.  yds 

14  sq.  rds.  10|-  sq.  yds.  3  sq.  ft.  38  sq.  in. 
=  14  sq.  rds.  10  sq.  yds.  7  sq.  ft.  110  sq.  in. 


Exercise  XXXIX. 


1.  Reduce  7  cu.  yds.  13  cu.  ft. 
to  cubic  feet. 

7  cu.  yds.  13  cu.  ft. 
X27 
189 
_13 
202  cu.  ft. 

2.  Reduce  25  cu.  yds.  5  cu,  ft, 
143  cu,  in.  to  cubic  inches. 

25  cu.  yds.  5  cu.  ft.  143  cu.  in. 

X27 

675 

5 

680  cu. 

X1728 


ft. 


1175040 
143 


1175183  cu.  in. 


3.  Reduce   74,325  cu.   in,  to 

cubic  feet, 

1728)74325  cu,  in. 

43  cu.  ft,  21  cu.  in, 
43  cu.  ft.  21  cu,  in, 

4.  Reduce  439,000  cu.  in.  to 
cubic  yards. 

1728)439000  cu,  in. 

27)254  cu.  ft.  88  cu,  in. 
9  cu.  yds,  11  cu.  ft, 
9  cu.  yds.  11  cu.  ft,  88  cu.  in. 

5.  Reduce  921,730  cu,  in,  to 
cubic  yards, 

1728)921730  cu.  in. 

27)533  cu.  ft.  706  cu.  in. 

19  cu.  yds.  20  cu.  ft. 

19  cu.  yds.  20  cu.  ft.  706  cu.  in. 


184 


ARITHMETIC. 


6.  Wood  cut  in  lengths  of 
4  ft.  is  piled  to  a  height  of  3J  ft. 
How  long  must  the  pile  be  to 
contain  a  cord  ? 

S^ft. 
Xj    ft. 

14  sq.  ft. 

9fft 
14)l28 
126 


Exercise 

1.  Reduce  3  pks.  5  qts.  1  pt. 
to  pints. 

3  pks.  5  qts.  1  pt. 
X8 

24 
5 

29  qts. 
X_2 

58 

1 

59  pts. 

2.  lieduce  4234  pts.  to  bushels. 

214234  pts. 


2117  qts. 


264  pks.  5  qts. 


7.  A  pile  of  wood  127  ft.  long, 
4  ft.  wide,  and  3  ft.  8  in.  high  is 
sold  for  $  7  a  cord.  How  much 
money  is  received  for  it  ? 

127  ft. 

X4ft. 

508  sq.  ft. 

X3f 

128)1862^ 

143|  cd. 
X$7 
flOlff  =  $101.86. 

XL. 

3.  Reduce  24  gals.  2  qte.  1  pt. 
2  gi.  to  gills. 

24  gal.  2  qts.  1  pt.  2  gi. 
X4 

96 
2 


98  qts. 
X2 

196 

I 

197  pts. 
Xj 

788 
2 


66  bu.  790  gi. 

66  bu.  5  qts. 

4.   Reduce  272  liquid  quarts  to  dry  quarta. 
17       11 

1       67i       1         4       W       4 


-  233=1  qts. 


TEACHERS     EDITION. 


185 


5.  'Reduce  400  dry  quarts  to  liquid  quarts. 
16  80 


m 

57| 


X400- 


11 


5120 
11 


465^5^  qts. 


6.  Express  a  bushel  in  cubic  feet,  carrying  the  decimal  to  three 
places. 

1.244 

1728)2150.420 

7.  Express  a  cubic  foot  as  the  decimal  fraction  of  a  bushel. 

0.8036 


215042)172800.0000 

8.   Reduce  1715|  bushels  to  pints. 

1715 

|bu. 

X64 

109792 

pts. 

9.   3047  gals,  to  barrels. 

3047     3047 

2 
X  — = 

=  ^094  =  96pbbl. 

3H         1 

63 

63           ^" 

Exercise  XLI. 

1.  Reduce  27,587  grs.  to  pounds 

3.    Reduce  136,851  oz.  to  tons. 

troy. 

16)136851  oz. 

24)27587  grs. 

100)8553  lbs.  3  oz. 

20)1149  dwts.  11  grs 

20]85  cwt.  53  lbs. 

12)57  oz.  9  dwt. 

4  t.  5  cwt. 

4  lbs.  9  oz. 

4  t.  5  cwt.  53  lbs.  3  oz. 

4  lbs.  9  oz.  9  dwt.  11  grs. 

4.   Reduce  864,205  grs.  (troy) 

2.   Reduce  34,652  lbs.  to 

ong 

to  pounds. 

tons. 

24)864205  grs. 

112)34652  lbs. 

20)36008  dwts.  13  grs. 

20)309  1.  cwt.  44  lbs. 

12)1880  oz.  8  dwts. 

151.  t.  9  1.  cwt. 

150  lbs. 

15  1.  t.  9  1.  cwt.  44  lbs. 

150  lbs.  8  dwts.  13  grs. 

186 


ARITHMETIC. 


6.   Reduce  864,205  grs.  (apoth.) 
to  pounds  avoirdupois. 

m^m  lbs. 
7000)864205 
123fH^  =  123T^lb8. 

-123  lbs.  7oz.  5.2dr8. 

6.  Reduce  5  lbs.  7  oz.  6  dwts. 
12  grs.  to  grains. 

51bs.  7oz.6dwtl2gr8. 
Xl2 
60 
_7 

67  oz. 
X20 
1340 

6 

1346  dwt. 
X24 
32304 

12 

32316  grs. 

7.  Reduce  745  lbs.  avoirdupois 
to  troy  weight. 

175 

»X^lb8. 
W0      1 
144 

=  905  lbs.  4  oz.  11  dwt.  16  grs. 

8.  Reduce    745    lbs.    troy    to 
avoirdupois  weight. 

144       149 

xm    1 

35 
-  613,>5  lbs.  =  613  lbs.  7^  drs. 


9.   Reduce  23,047,125  drs.  to 

tons. 

16)23047125  drs. 
16)1440445  oz.  5  drs. 
100)90027  lbs.  13  oz. 
20)900  cwt.  27  lbs. 
45  t. 

45  t.  27  lbs.  13  oz.  5  drs. 


10.   Reduce  90,252,381  drs.  to 
tons. 

16)90252381  drs. 
16)5640773  oz.  13  drs. 
100)332548  lbs.  5  oz. 
20)3525  cwt.  48  lbs. 
176  t.  5  cwt. 

176  t.  5  cwt.  48  lbs.  5  oz.  13  drs. 


11.  Reduce  1  pint  to  minims. 
1  fl.  oz.  xvj. 
16 

16  fl.  drm.  viij. 
8 

128  nt  Ix. 
60 

7680  nt. 


12.   Reduce  8000  jti  to  ounces. 

60)8000  ni, 

8)135  m  Ix.  20  m- 

16  fl.  drm.  viij.  5  n^  l^- 

16  fl.  drm.  viij.  5  rrj,  Ix.  20  n^^. 


TEACHERS     EDITION. 


187 


Exercise  XLII. 


1.   Reduce  6  hrs.  17  min.  25 

5.   F 

md  the  number  of  days, 

sec.  to  seconds. 

reckoning  from  noon  of  the  one 

6  hrs.  17  min.  25  sec. 

to  noon 

of  the  other,  between  the 

X60 

following  days  in  the  year  1880: 

360 

July  4  and  December  2 

Febru- 

17 

ary  1  and  May  29;   January  3 

377  min. 
X  60 

and  October  15;    also, 

between 

December  25.  1880  and 

May  25, 

22620 

1881. 

25 

22645  sec. 

27  d. 

28  d.        28  d. 

6d. 

31  d. 

31  d.        29  d. 

31  d. 

2.   Reduce  1  yr.  13  dys.  4  min. 
to  minutes. 

30  d. 

30  d.        31  d. 

28  d. 

1  yr.  13  d.  4  min. 

31  d. 

29  d.        30  d. 

31  d. 

X365 

30  d. 

118  d.        ^^  ^• 

30  d. 

365 

2d. 

30  d. 

25  d. 

13 

151  d. 

31  d. 

151  d. 

378  d. 

aid. 

X24 

30  d. 

9072  h. 

15  d. 

X60 

544320 
4 
544324  min. 

286  d. 

6.   How    many    miautes    are 

3.   Reduce  48,567  min.  to  days. 

there  from  midnight  of  March  7 

60)48567  min. 

to  midnight  of  June  20  ? 

24)809  hrs.  27  min. 

24  d. 

33  d.  17  hrs. 

30  d. 

33  d.  17  hrs.  27  min. 

31  d. 
20  d. 

4.   Reduce  742,392sec.  to  days. 

105  d. 

60)742392  sec. 

X  24 

60)12373  min.  12  sec. 

24)206  hrs.  13  min. 

2520  hrs. 

8  d.  14  hrs. 

X60 

8  d.  14  hrs.  13  min.  12  sec. 

151200  min. 

188 

ARITHMETIC. 

7.   Find  the  number  of  seconds 

8.  Which  of  the 

years   1600, 

from  eight  o'clock  Monday 

morn- 

1656,    1700,    1734, 

1800,    1818, 

ing  till  six  o'clock  the  next  Sat- 

1880, 1900,  1924,  2000  are  leap 

urday  evening. 

• 

years  ? 

16  hrs. 
24  hrs. 
24  hrs. 
24  hrs. 

1600  (divisible  by  400). 
1656          "         •'       4). 
1880          "         "       4). 

24  hrs. 

1924 

"       4). 

18  hrs. 

2000 

"  400). 

130  hrs. 

X60 

7800  min. 

X60 

468000  sec. 

Exercise  XLIII. 


1.  Reduce  2°  30^  25^^  to  seconds. 

2°  30'  25^' 
X60 
120 
_30 
150' 
X60 
9000 
25 
9025'' 

2.  Reduce  15°  3'  22"  to  seconds. 

15°  3'  22". 

X_60 

900 

_3 

903' 

X60 

54180 

22 

54202'^ 


3.   Reduce  56,760"  to  degrees. 

60)56760"  ■ 
60)946' 
15°  46^ 


4.    Reduce  212,221"  to  degrees. 

60)2^2221" 
60)3537'  1" 
58°  57' 
58°  57'  1". 


6.  The  hour  and  minute  handa 
of  a  watch  form  an  angle  of  how 
many  degrees  at  3  o'clock?  at 
4  o'clock?  at  6  o'clock?  at  7^ 
o'clock?  at  11  o'clock?  at  12 
o'clock  ? 


TEACHERS     EDITION. 


189 


12     ;;2      2      8 
4 


45^ 


t\  =  1  =  120°. 


0°. 


6.   How  many  geographical  miles  in  the  width  of  the  torrid  zone 
(46°  550  ?     How  many  statute  miles  ? 

^         46°  55^  46°  55^  =  46|f  °  =  46.91|. 


2760 
55 

2815^ 
=  2815  geog.  mi. 


46.911 
X  69.16 

3244.75661 

=  3244.751  Stat.  mi. 


Exercise  XLIV. 


1.  Reduce  £583  6  s.  8  d.   to 
pence. 

£583  6  s.  8  d 
X20 


11660 


11666  8. 
Xl2 
139992 


140000  d. 

2.   Reduce  £79  18  s.  11^  d.  to 
farthings. 

£79  18  s.  ll^d. 
X20 
1580 
18 
1598  s. 
Xl2 


19176 


19187^  d. 

X4 
76750  far. 


3.   Reduce  28,572  d.  to  pounds. 

12)28572  d. 
20)2381  s. 
£119  18. 


4.  Reduce  272,191  d  to  half- 
sovereigns. 

12)272191  d. 
10)22682  s.  7  d. 

2268  half-sov.  2  s.  ' 
2268  half-sov.  2  s.  7  c?. 


5.   Reduce   27,281   crowns   to 
guineas. 

27281  half-crowns. 
^5 


21)136405  s. 

6495^.  10  5. 


190                                             ARITHMETIC. 

6.  Reduce   1,716,114   guineas 

9.   Reduce     286,347     far.    to 

to  pounds. 

1716114  ^r. 
X21 

crowns. 

4)286347  far. 
12)71586  d  3  far. 
5)5965  8.  6d. 

20)36038394  «. 
£1801919  14  s. 

1193  crowns. 
1193  crowns  6 d3far. 

7.  Reduce    291,374    far. 

to 

10.   Reduce  20  francs  to  dollars. 

pounds. 

$0,193 

4)291374  far. 
12)72843  (f.  2  far. 
20)6070  8.  3d 
£303  108. 

X20 

$3.86 

11.  Reduce  20  marks  to  dollars. 

£303  108.  3d  2  far. 

$0,238 
X20 

8.  Reduce  709,126d  to  guineas. 

$4.76 

12)709126  d 

12.  Reduce  5  roubles  to  dollars. 

21)59093  8.  10  d 
2813^.  20  8. 

$0,734 
X5 

2813^.  20  8.  10  d 

$3.67 

13.   Find  the  whole  sum  of  money  in  a  box  containing  35  sover- 
eigns, 27  half-sovereigns,  13  crowns,  41  half-crowns,  and  85  shillings. 
35  sovereigns    =700s.  20)1222^8. 

27half-8ov.       =2708.  £612^-8. 

13  crowns  =    65  a. 

41  half-crowns  =  102  J  «. 
85  shillings       -    858. 


=.£61  28.  6d 


1222^8. 

Exercise  XLV. 
1.  Express  59°  F.  in  Centigrade  scale  ;  in  Reaumur's  scale. 
59°  F.  is  27°  above  freezing-point. 


3 

19 


15°  C. 


3 
19 


12°  R. 


teachers'  edition.  191         CZ 


2.  Express  77°  F.  in  Centigrade  scale ;  in  Reaumur's  scale. 

77°  F.  =  45°  above  freezing-point. 

5  5 

^  X  -  C.  =  25°  C.  -  ^  X  -  R.  =  20°  R. 

1      ^  IF 

3.  Express  950°  F.  in  Centigrade  scale ;  in  Reaumur's  scale. 

950°  F.  =  918°  above  freezing-point. 
102  102 

®  X  -  C.  =  510°  C.  &x-B..  =  408°  R. 

4.  Express  —  40°  F.  in  Centigrade  scale  ;  in  Reaumur's  scale. 

—  40°  F.  =  72°  below  freezing-point, 
f  of-  72°  C.  =  -  40°  C  f  of-  72°  R.  =  -  320°  R. 

5.  Express  —  4°  F.  in  Centigrade  scale  ;  in  Reaumur's  scale. 

—  4°  F.  =  36°  below  freezing-point. 
^  of-  36°  C.  =  -  20°  C.  I  of-  36°  R.  =  -  16°  R. 

6.  Express  10°  C.  in  Fahrenheit's  scale ;  in  Reaumur's  scale. 


7.  Express  22°  C.  in  Fahrenheit's  scale ;  in  Reaumur's  scale. 
22°  C.  =  I  of  22°  +  32°  F.  =  71f°  F.        f  of  22°  R.  =  17f°  R. 

8.  Express  —  30°  C.  in  Fahrenheit's  scale  ;  in  Reaumur's  scale. 

-  30°  C.  =  f  X  -  30°  +  32°  F.  =  -  22°  F.       f  x  -  30°  R.  =  -  24°  R. 

9.  Express  —  llf°  C.  in  Fahrenheit's  scale ;  in  Reaumur's  scale. 

-  llf°  C.  ==  f  X  -  llf°  +  32°  F.  =  llf°  F.       I X  -  llf°  R.  =  -  9|°  R. 

10.  Express  4°  R.  in  Fahrenheit's  scale  ;  in  Centigrade  scale. 
4°  R.  -  f  of  4°  +  32°  F.  =  41°  F.  f  of  4°  C.  =  5°  C. 

11.  Express  12°  R.  in  Fahrenheit's  scale  ;  in  Centigrade  scale. 
12°  R.  =  f  of  12°  +  32°  F.  =  59°  F.        f  of  12°  C.  =  15°  C. 


192 


ARITHMETIC. 


12.  Express  —  20°  R.  in  Fahrenheit's  scale  ;  in  Centigrade  scale. 
20°  R.  =  f  of- 20°  +  32°  F.  =  -13°  F.        ^  of-20°  C.  =  -25°  C. 

13.  Express  4°  C.  in  Fahrenheit's  scale ;  in  Reaumur's  scale. 
4°  C.  =  f  of  4°  +  32°  F.  =  39^°  F.  f  of  4°  R.  =  3|°  R. 

14.  Express  0°  F.  in  Centigrade  scale  ;  in  R6aumur's  scale. 

0°  F.  =.  32°  below  freezing-point. 
^  of-  32°  C.  =  -  17^°  C.  I  of-  32°  R.  -  -  14f°  R. 


Exercise  XLVI. 


1.  Add: 

hn. 
14 
17 

9 
12 
22 


mln. 

21 
13 

47 
53 
17 


37 
32 

43 
54 
50 


3dys.  4 

34 

36 

2.  Add: 

on.  ydi. 

ou.  ft. 

cu.  In. 

130 

5 

820 

56 

20 

304 

37 

4 

86 

8 

10 

129 

12 

19 

175 

245 

4 

1514 

3.  Add: 

t. 

■. 

d. 

35 

2 

6f 

18 

6 

4 

27 

3 

10 

12 

0 

5 

12 


1  3  far. 


4.  Add: 


ml. 

rd«. 

jiB. 

ft. 

iB 

6 

120 

3 

2 

2 

18 

15 

1 

1 

6 

3 

215 

2 

2 

3 

7 

95 

1 

1 

8 

35 

126 

3i 

1 

7 

h  = 

1 

6 

35   126 


5.  Add: 

A.  8q.  rds.  iq.  yds.  sq.ft.  sq.  in, 

74  21  5    4  100 

23  37  13    5  83 

12  106  17    8  7 

41  50  23    0  24 


151   55 


29J   0 


151   55 


29 


70 
108 


34 


6.  Add  5  bu.  3  pks.  6  qts.  1  pt. ; 
6  bu.  2  pks.  7  qta. ;  7  bu.  1  pk. 


TEACHERS     EDITION. 


193 


1  qt.  1  pt. ;  1  pk.  7  qts. ;  2  bu.  3 


pks.  1  pt. 

bu. 

5 
6 

7 
0 
2 


pks. 

3 
2 
1 
1 
3 


23 


0 


6 


7.  Add  48  t.  13  cwt.  75  lbs.  6 
oz.  4  drms. ;  25  t.  12  cwt.  27  lbs. 
8  oz.  13  drms. ;  51  t.  10  cwt.  44 
lbs.  15  drms. ;  80  t.  5  cwt.  6  oz. ; 
19  cwt.  27  lbs. ;  25  lbs.  8  oz.  10 
drms. ;  5  t.  5  cwt.  5  oz. 

t.         cwt.  lbs.  oz.  drs. 

48  13  75  6  4 

25  12  27  8  13 

51  10  44  0  15 

80  5  0  6  0 

0  19  27  0  0 

0  0  25  8  10 

5  5  0  5  0 


212        6 


10 


&■■■ 
gall 
qts. 
pts. 
gi- 


8.  Add  50  gals.  3  qts.  1  pt.  3 
12  gal.  1  qt.  1  pt.  1  gi. ;  5 
.  2  qts.  1  pt.  2  gi. ;  75  gal.  3 
1  pt.  3  gi. ;  80  gals.  3  qts.  0 
1  gi. ;  17  gals.  1  qt.  1  pt.  3 


50 
12 
5 
75 
80 
17 
243 


9.  Add  13  lbs.  4  oz.  8  dwt.  6 
grs. ;  25  lbs.  8  oz.  13  dwt.  20  grs. ; 
8  lbs.  11  oz.  14  grs. ;  20  lbs.  16 
dwt.  8  grs. ;  15  lbs.  9  oz.  12  dwt. ; 
4  oz.  3  dwt. 


13 
25 

8 
20 
15 

0 


11 
0 


13 

0 

16 

12 

3 


20 

14 

8 

0 

0 


84 


14 


10. 

3  gals. 

qts. ;  14  gals.  1^  pts. 

qts.  1  pt. 


Add  4  gals.  3  qts.  1  pt. ; 

2  qts.  li  pts.;  12  gals.  3 

5  gals.  2 


gals. 

4 

3 

12' 
14 

5 


41 


J 

n 

0 

U 
1 

1 


11.  Add  60°  50^  50^/ ;  20°  41^ 
52^^ ;  30°  25^  20'^  ;  20°  32^  43^^. 


60 

50 

50 

20 

41 

52 

30 

25 

20 

20 

32 

43 

132 


30 


45 


194 


ARITHMETIC. 


Exercise  XLVII. 


1.  Subtract  23  lbs.  8  oz.  19 
dwt.  10  grs.  from  58  lbs.  6  oz.  17 
dwt.  21  grs. 

lb«.  oz.  dwt.  grs. 

58        6  17         21 

23        8  19  10 


34 


18 


11 


2.  Subtract  5  bu.  I  pk.  n  qts.  1 
pt.  from  5  bu,  3  pks.  3  qts. 

bn.  pka. 

5  3 

5  1 


qU. 

3 


3.  Subtract  32  cu.  yds.  13  cu. 
ft.  1600  cu.  in.  from  39  cu.  yds. 
17  cu.  ft.  1400  cu.  in. 

on.  ydf .  ca.  ft.  cu.  in. 


39 
32 


1400 
1600 


7  3  1528 

4.  Subtract  £92   15  «.    1^  d. 
from  £120  13  «.  4  d 


£. 

120 
92 


•.  d. 

13  4 

15  1^ 


27 

18 

2 

3  far. 

5.  Subtract  22  gals 
from  30  gals.  2  qta. 

3  qta 

Ipt. 

gala. 

30 

qu. 
2 

pt.. 
0 

22 

3 

1 

6.  Subtract  17 1.  7  cwt.  17  lbs. 
6  oz.  10  drs.  from  25  t.  13  cwt. 
15  lbs.  12  oz.  5  drs. 

t.        cwt.         lbs. 

25      13        15 
17       7       17 


drs. 

5 

10 


98 


11 


7.  Subtract  13  A.  150  sq.  rds. 
98  sq.  ft.  10  sq.  in.  from  20  A. 

A.  sq.  rds.  sq.  ft.  sq.  in. 

20  0  0  0 

13       150         98         10 

"e          9       173^      134 
i=    36 

6  9       174         26 

8.  Subtract  58°  33'  36'^  from 
90°  11'  21''. 


90 
58 


11 
33 


21 
36 


31 


37 


45 


9.   Subtract  2  yrs.  213  dys.  17 
hrs.  from  3  yrs.  147  dys.  14  hrs. 

yrs.  dys.  hrs. 

3  147  14 

2  213  17 


298  21 

10.  Subtract  3  mi.  217  rds.  4 
yda.  1  ft.  3  in.  from  4  mi.  100  rds. 
3  yds. 

ml.         rds.         yds.  ft.  In. 

4       100       3         0  0 

3       217       4         1  3 


202       3i       1 
202      4        0 


TEACHERS     EDITION. 


195 


Exercise  XLVIII. 


1.   Multiply  £31  2  s.  6^  d.  by 


31 


249  0  4 

2.   Multiply  19  gals.  3  qts.  1 
pt.  by  70. 

gals.  qts.  pts. 

19  3  1 

70 


1391  1 


0 


3.  Multiply  3  lbs.  4  oz.  8  dwt. 
10  grs.  by  10. 

lbs.  oz.  dwt.  grs. 

3  4  8  10 

10 

33        8  4  4 

4.  Multiply  5  t.  10  cwt.   67 
lbs.  by  10. 

t.  cwt.  lbs. 

5  10  67 

10 


55  6  70 

5.   Multiply  43  bu.  2  pks.  by 


G3 


ba. 
43 


304 


2740 


pks. 

2 


6.   Multiply  15  wks.  3  dys.  5 
hrs.  12  min.  by  7. 

wks.  dys.  hrs.  min. 

15  3  5  12 

7 

108  1         12         24 


7.   Multiply  5  cu.  yds.  IG  oa. 
ft.  371  cu.  in.  by  6. 

cu.  yds.  cu.  ft.  cu.  in. 

5  10  371 

6 


32 


498 


8.   Multiply  27  A.  76  sq   ids. 
22  sq.  yds.  5  sq.  ft.  by  90. 

A.     sq.  rds.   sq.  yds.  sq.  ft. 

27      76      22      5 
9 


247 

50 

21|  0 

247 

50 

21   4| 
10 

3   553 

27 

^      0   sq.in 

\=2      36 

3      553      27        3      2      36 


196  ARITHMETIC. 


9.   Multiply  32  rds,  3  yds.  1  ft.  by  57. 

57                                 57  57 

Xj                                  X3  X32 

3)57                              171  1824 

19  yds.                        J9  _34 

5^) 190  320) 1858 

_2  5  mi.  258  rds. 
11)380 

34  ...  6  half-yds.  =  3  yds. 

5  mi.  258  rds.  3  yds.  Ana. 


10.  Multiply  34  dy8.  10  hrs.  13  min.  12  sec.  by  108. 

108  108 

X  12  X  13 


60 


1296  1404 

216  _11 


21  .  .  .  T%  min.  =  36  sec.  60)1425 

23  ...  45  min. 


108  108 

XlO  X34 

1080  3672 

23  45 

24)1103  365)3717 

45  ...  23  hrs.  10  .  .  .  67  dyi 

10  yrs.  67  dys.  23  hrs.  45  min.  36  sec.  Am. 
11.   Multiply  5  mi.  126  rds.  19  yds.  6  in.  by  7125. 


7125 
X6 

.  6  in. 
.1ft. 

7125 
X  19 

2 

42750 

135375 

3 

3662. 

1187 

1187  .  . 

5i)  136562 

2 

11)273124 

24829 

|-2iyd8. 


TEACHERS     EDITION. 


197 


7125 
X126 

320)897750 
2883 


19  rds. 


7125 
X5 

35625 

2883 


38508  mi. 
38,508  mi.  19  rds.  2|-  yds.  1  ft.  6  in.  =  38,508  mi.  19  rds.  3  yds.  Ans. 


12.  Multiply  11  5  5  32  3  11  grs.  by  2197. 


2197 
Xll 

20) 24167 
1208 


2197 
X5 

10985 

1867 

8)12852 

1606 


7  grs. 


13. 


43 


2197 
X2 

4394 

1208 

3) 5602 

1867 

2197 
Xll 

24167 
1606 

12)25773 

2147  lbs.  9  5. 


2147  lbs.  9543137  grs.  Ans. 


Exercise   XLIX. 


1.   Divide  54  mi.  124  rds.  1  yd.  2  ft.  6  in.  by  33. 


33)54 

rd8. 
124 

1- 

ft. 

2 

in. 
6 

(1 

33 

33)724(2 

21 

66 

X320 

6i 

6720 

X3 

124 

2U 

33)6844(207 

Xl2 

66 

258 

244 

6 

231 

33)264(8 

13 

264 

X_5i 

72^ 

1 

mi, 

,  207  rds.  2  yds. 

198 


ARITHMETIC. 


2. 

Divide  5  cu. 

yds.  1 

cu 

84  cu 

.  in 

by  1716 

cu.  in. 

on.  yd*,    cu.  ft. 
5            1 

X27 
135 

1 

cu. 

S4 

136 

X  1728 

235008 
84 

1716)235092(137.  Ans. 

3.   Divide   8426  wks.    6  dys. 
21  hrs.  10  min.  21  sec.  by  1029. 

wka.      Ay:   hri.     min.     sec. 

1029)8426  6  21  10  21(8 
8232 
194 

x7(+6) 

1029)1364(1 

1029 

335 

X24 

8040 

21 

1029)8061(7 
7203 
858 
X60 

1029)51490(50 
5145 

40 

60 

2400 

21 

1029)  2421  (2TVW=-2iH- 
2058 

363 

8  wks.  Idy.  7  hrs.  50  min.  2|f^8ec. 

Ana. 


4.   Divide  £394  2  8.  10^  d.  by 
£5  28.  4H- 

£394  2  s.  10^  d.  =  378,378  far. 
£5  2  8.  4^  d  =  4914  far. 

77.  Ans. 


4914)378378 

5.   Divide  22  wks.  2  dys.  by 
11  hrs.  31  min.  12  sec. 

22  wks.  2  dys.  =  13,478,400  sec' 
llhr8.31min.  12sec.=41,4728ec. 
325.  ^718. 


41472)13478400 

6.   Divide   74,128  sq.  mi.  517 
A.  80  sq.  rds.  by  10,000. 


Bq.  mi. 

10000)74128 

70000 

4128 

X640 


A.   sq.  rda. 

517  80(7 


2641920 
517 


10000)2642437(264 
2640000 

2437 

Xl60 

389920 

80 

10000)390000(39 
390000 

7  sq.  mi.  264  A.  39  sq.  rds.  A 


71  s. 


7.  Divide  38°  37'  42'^  by  5*= 
3V  6'^ 

38°  37'  42''  =  139,062". 
5°  31'  6"  =  19,866". 

19866)139062(7.  Ans. 
139062 


TEACHERS     EDITION. 


199 


Exercise  L. 


1.  Find  the  value  of  f  of  a 
mile. 

I  mi.  =  f  of  320  rds.  =  256  rds. 
Ans. 

2.  Find  the  value  of  j\  of  an 
acre. 

j\  A.  =  j\  of  160  sq.  rds. 
=  30  sq.  rds.  Ans. 


3.  Find   the  value   of  f  of  a 
hundredweight. 

f  cwt.  =  f  of  100  lbs.  =  62i  lbs. 
^  lb.  =  I  of  16  oz.  =  8  oz. 
62  lbs.  8  oz.  Ans. 

4.  Find   the   value  of  f  of  a 
pound  sterling. 

£f-f  of  20s.  =  13|s. 

is.  =  iof  12d  =  4d 

13  s.  4  d.  Ans. 


5.  Find  the  value  of  -^j  of  a  mile. 

t\  mi.  -  j\  of  320  rds.  -  261^^1  rds. 
t\  rds.  =  /t  of  51  yds.  =  41  yds. 
i  yd.  =  1  of  3  ft.  =  11  ft. 

^  ft.  =  1  of  12  in.        =  6  in. 

261  rds.  4  yds.  1  ft.  6  in.  Ans. 

6.  Find  the  value  of  -^j  of  an  acre. 

TT  A.  =  i?T  of  160  sq.  rds.  =  101/^-  sq.  rds. 
^  sq.  rds.  =  j\  of  30^  sq.  yds.  =  24|  sq.  yds. 
f  sq.  yds.  =  |  of  9  sq.  ft.  =  6f  sq.  ft. 

f  sq.  ft.  =  f  of  144  sq.  in.     =  108  sq.  in. 

101  sq.  rds.  24  sq.  yds.  6  sq.  ft.  108  sq.  in.  Ans. 


7.   Find   the   value  of  f  of  a 
degree. 

f°  =  |of60^  =26f^ 

Y  =  I  of  60^^  =  40^^. 

26'  W.  Ans. 


8.   Find   the  value  of  |^  of  a 
year. 

I  yr.  =  ^  of  365  dys.  =  121|  dys. 
f  dy.  =  I  of  24  hrs.    =  16  hrs. 
121  dys.  16  hrs.  Ans. 


200 


ARITHMETIC. 


9.  Find  the  value  of  0.15625 
of  a  bushel. 

0.15625  bu. 

Xj 

0.625  pks. 

X8 
5  qts.  Am. 

10.  Find  the  value  of  0.625  of 
a  gallon. 

0.625  gal. 

X4 
2.5  qte. 
X2 
Ipt. 
2  qts.  1  pt.  Ans. 


11.   Find  the  value  of  0.875  of 
a  leap-year. 

0.875 
X366 


320.25  dys. 
X24 


6hr8. 
320  dys.  6  hrs. 

12.   Find  the  value  of  0.325  of 
a  pound  troy. 

0.325  lbs. 
Xl2 
3.9  oz. 
X20 
18  dwt. 
3  oz.  18  dwt.  Ana. 


1.   Find  the  value  of 


Exercise  LI. 

f  of  3  A.  101^  sq.  rds. 


sq.  rdi. 

lOli 
X2^ 


5)6  202f 

1  72     16  sq.  yds.  1  sq.  ft.  28|  sq.  in. 

A.  sq.  rds. 

3  lOH 

X6 


21 
1 


128 

72 


•q.  yd«. 
16 


■q.  ft. 

1 


sq.  in. 

28^ 


23 


40 


16 


2.   Find  the  value  of  If  of  7 
hrs.  21  min.  27  sec. 


7 

21 

27 
X3 

7)22 

4 

21 

3 

7 

9 
21 

11^ 
27 

10 


30 


38^ 


1  28t 

3.  Find  the  value  of  10.0175  of 
1  dy.  13  hrs. 

10.0175 
X  37  hrs. 
370.6475 
X60 
38.85  min. 
X60 
51  sec. 
370  hrs.  38  min.  51  sec.  =- 15  dys. 
10  hrs.  38  min.  51  seu.  Ant, 


TEACHEES     EDITION. 


201 


4.   Find  the  value  of  17tV  of 
10  ydb.  2  ft.  31  in. 

yds.  ft.  in. 


10 

2 

x7 

12)75 

0 

m 

6 

0 

9il 

10 

ft. 
2 
X 

in. 

1?^ 

182 
6 

2 
0 

6f 
9H 

189 

0 

4t\ 

34  rds.  2  yds.  0  ft.  ij%  in.  Ans. 

5. 

Find  the  value 

of  0.01284 

of  14 

mi. 

0.01284 
X  14  mi. 

0.17976  ] 
X320 

Tii. 

57.5232  rds. 
X5i 
2.8776  yds. 
X3 

2.6328  ft. 
X  12 

7.5936  in. 
57  rds.  2  yds.  2  ft.  7.5936  m.  Ans. 

6.    Find  the  value  of  0.42776 
of  12  t.  10  cwt. 

10  cwt.  =  ^  t. 

0.42776 
Xl21t. 
5.347  t. 
X20 
6.94  cwt. 

xioo 

94  lbs. 
5  t.  6  cwt.  94  lbs.  Ans. 


+  31  oz.  +  5 1  dwt. 


4|  oz.  +  3|  oz.  =  8^  oz. 
J^  oz.  -  ^V  of  20  dwt.  =  f  dwt. 
f  dwt.  +  5f  dwt.  =  6i  dwt. 
^  dwt.  =  ^  of  24  grs.  =  2f  grs. 

8  oz.  6  dwt.  2|  grs.  Ans. 


8.   Find  the  value  of  0.35  of 
4  lbs.  5  oz.  6  dwt.  16  grs. 

lbs,  oz.         dwt.         grs. 


16 

X7 


20)31 


13 


16 


8  Ans. 


9.   Find    the   value    of   3.726 
mi.  —  33.57  rds. 

3.726  mi. 
X320 


1192.32  rds. 

33.57 
1158.75  rds. 
=  3  mi.  198  rds.  4  yds.  4^  in.  Ans. 

10.  Find  the  value  of  y\  of  a 
year  +  -^-^  of  a  week  +  ^j  of  an 
hour. 

■^\  yr.   =  7^  of  365  dys.  =  15  dys. 
Awk.  =  ^^of     7  dys.  =lidys. 

I  dy.  =    ^  of    24  hrs.  =  3  hrs. 
x^^jhr.   =/jof    60  min.  =  35  min. 


15  dys. 
Idy. 


3  hrs. 


35  min. 


16  dys.       3  hrs.        35  min. 
2  wks.  2  dys.  3  hrs.  35  min.  Ans. 


202 


AEITHMETIC. 


11.   Find  the  value  of  5.268  of  2  dys.  +  2.829  of  16  hrs.  +  0.9528 
of  25  min. 


6.268 
X  2  dys. 


2.829 
X  16  hrs. 


10.536  dys. 
X24 


45.264  hrs. 
12.864  hrs. 


0.9528 
X  25  min. 

23.82  min. 
7.68  min. 


12.864  hrs. 


31.5    min. 
X60 


30  sec. 


58.128  hrs. 
X60 

7.68    min. 

10  dys.  58  hrs.  31  min.  30  sec.  =  12  dys.  10  hrs.  31  min.  30  sec.  Arts. 

12.   Find  the  value  of  ^^  of  a  mile  +  f  of  40  rds.  +  f  of  a  yd. 

^5  mi.  =  j%  of  320  rds.  =  60  rds. 
f  of  40  rds.  =  26|  rds. 


60  rds.  +  262  rds. 
I  rds.  =  I  of  5^  yds 

f  yd.  +  f  yd. 

1^^  yds.  +  3  yds. 

86  rds.  4  yds.  1^  in.  Ans. 


=  86 1  rds. 
=  3 1  yds. 
=  l,Vyd8. 

=  4^Vyd8. 


13.    Find  the  value  of  f  of  2  cwt.  84  lbs.  +  f  of  5  cwt.  98  lbs.  +  f 
of  7i  lbs. 


2  cwt.  84  lbs. 
XlOO 

4)284  lbs. 

71 

X3 

100)213  lbs. 

2  cwt.  13  lbs. 


5  cwt.  98  lbs. 
XlOO 


698  lbs. 
X3 


7 
100 


1794 


25 


56^  lbs. 
2  cwt.  56  lbs.  4  oz.  9}  drs. 


f  of  7i  lbs.  =  3  lbs. 
2  cwt.  13  lbs. 
2  cwt.  56  lbs.  4  oz.  9f  drs. 
3  lbs. 

4  cwt.  72  lbs.  4  oz.  d\  drs.  Am. 


teachers'  edition.  203 


Exercise   LII. 

1.  Express  a  pound  avoirdupois  as  the  fraction  of  a  pound  troy. 

1  lb.  avoird.  =  7000  troy  grp. 
1  lb.  troy       =  5760  troy  grs. 

5760      144 

2.  Express  an  ounce  avoirdupois  as  the  fraction  of  an  ounce  troy. 

437-1  grs. 
16)7000  1  oz.  troy  =  480  grs. 

437^^875^175   ^^ 
480      960      192 

3.  Express  363  sq.  yds.  as  the  fraction  of  an  acre. 

Xl60  4840     40 

160  sq.  rds. 
X  30| 

4840  sq.  yds. 

4.  Express  ^  oi  £2  I  s.  3d.  +  j\  of  £1  As.  9d.  as  the  fraction  of 
£2  14s. 


14 


£     «. 

2    1 

d. 
3 

£ 
1 

4 

d. 

9 

£ 

2 

X20 

X20 

X20 

41  8. 

24  s. 

54  s. 

Xl2 

5)495d 
99 

Xl2 

11^297  d. 

27 

X  12 
648  cZ. 

X3 

297  d 

X5 
135^. 

297 

+  135   432 

_2 

Ans. 

672  648     3 


204  ARITHMETIC. 


5.   Express  2  mi.  138  rds.  1  yd.  as  the  fraction  of  3  mi.  265  rds. 
3  yds.  1  ft.  6  in. 

2  mi.  138  rds.  1  yd.  3  mi.  265  rds.  3  yds.  1  ft.  6  in. 

X320  X320 

640  960 

138  265 

778  rds.  1225  rds. 

X5^  X5^ 

4280  yds.  6740|  yds. 

X36  X  3 


154080  in.  20222^  ft. 

1^4080^40^^^^  24liin. 

242676     63 

6.   Express  f  of  560  lbs.  as  the  fraction  of  5  long  tons. 
I  of  560  lbs.  =  160  lbs. 
2240  lbs.  160        1 


X  5  11200     70 


Ans. 


11200  lbs. 


7.  Express  f  of  200  rds.  as  the  fraction  of  4  miles. 

I  of  200  rds.  =  133J  rds. 

320  rds.  5 

X4  133±  =  _J_x^  =  A.  ^n«. 

— -  1280     Xm       3       48 

1280  rds.  16 

8.  Express  ^^  of  2  dys.  2  hrs.  24  min.  as  the  fraction  of  2  wks.  1  d. 

2  dys.  2  hrs.  24  min.  2  wks.  1  dy. 

X24  X_7 

60  hrs.  15  dys. 

X60  X_24 
27)3024  nvin.  360  hrs. 

112  x60 

X  10  21600  min. 


1120  min. 


21600     135 


teachers'  edition.  205 

9.   Express  f  of  the  difference  between  3  yds.  2  ft.  11  in.  and  10 
yds.  7  in.  as  the  fraction  of  8  yds. 


yds. 

10 

ft. 
0 

in. 
7 

3 

2 

11 

6 

0 

8 

X36 
224  in. 

8  yds. 
X36 

X4 

288  in. 

5)896 
179^  in. 

179^       1 

288    m 

28 
5 

_28 
45'  ■ 

Am. 

10.   Express  ^  of  the  difference  between  f  of  7  hrs.  and  ^  of  15 
min.  as  the  fraction  of  12  hrs.  18  min. 

7  hrs.  =  420  min.  12  hrs.  18  min. 


105 


60 


5     1^^525.  733  ^i^^ 

^       I  2 


2 


lx^  =  —  ^  525^21^2583 


3 

15'  2   "  5        10 


5 

123 

10     2382     -loo     •  123     1      . 

ML  X  tiTJL  =  123  mm.  — -  =  -.  Ans. 

^X       X^  738     6 

11.   Express  f  pt.  as  the  fraction  of  a  gallon. 

lgal.  =  8pte.  1  =  1x2  =  1.  ^«., 

4 


206  ARITHMETIC. 


12.   What  part  of  4  lbs.  1  oz. 

8  dwt.  15  grs.  is  1  lb. 

1  oz.  9  dwt. 

15  grs. 

4  lbs.  1  oz.  8  dwt.  15  grs. : 

1  ib.  1  oz.  9  dwt. 

15  grs.: 

15grs.  =^|dwt.    =|dwt. 

15  grs.  =  ^1  dwt. 

=  i  dwt. 

8fdwt.=|oz.      =,fi^oz. 

9f  dwt.  =^oz. 

=  l'«VOZ. 

WTrOZ.=^lb8.  =  ^^^lb8. 

li^oz.=iA^lb8, 

.  =  /^lbB. 

4^Viy  lbs. 

V^lbs. 

=n-^-- 

11 


13.   What  part  of  2  mi.  is  |  of  6  rds.  3  yds.  2  in.  ? 
6  rds.  3  yds.  2  in.: 

SI  yds.  =  ^  rds.  =  I  rd. 

6|  rds. 
I  of  6|  rds.  =  W  rds. 


iMs    lof 

2        2 

mi.  = 

59 
4320 

59 
8640" 

i. 
Ana. 

'..  What 

part 

of  a  Dushel  is  1  pk. 
1  pk.  2  qts. 
1  pt.  =  i  qt- 

2  qts 
Ipt.: 

.  Ipt.? 

2iqt8.=|pk8.= 

=  1^.^ 

l^a. 

l^  pks.  =  lA  bu.  =  f  i  bu.  Ans, 

15.  What  part  of  20  A.  are  19  A.  3.5  sq.  ch. 
19.  A.  3.6  eq.  ch. : 

3.5  8q.ch.-MA.  =  AA. 

1M  =  387     1^387   ^^ 
20        20  ^  20     400"     ^ 


TEACHERS     EDITION. 


207 


16.   What  part  of  5  tons  are  3  t.  240  lbs.  ? 
3  t.  240  lbs. : 
2401bs.  =  ^%V7t.  =  ^3^t. 


3&  =  lx^ 
5       5     25 


.  Ans. 

125 


17.   38  sq.  rds.  194  sq.  ft.  108  sq.  in  =  what  part  of  an  acre? 
38  sq.  rds.  194  sq.  ft.  108  sq.  in. : 
108  sq.  in.  =  ^f  f  sq.  ft.      =  f  sq.  ft. 
194^ 


194|  sq.  ft.  =  iHll  sq.  rds.  =  ^y/^  sq.  rds. 


3835,V^sq.rds.  =  ^^A.  = 


42161 
174240 


A.  Ans. 


Exercise  LIII. 


1.  Express  16  s.  3|d 
decimal  of  a  pound. 


the 


3.75  d. 
16.3125  s. 


£0.815625.  Ans. 

2.  Express  233  rds.  9  ft.  10.8  in. 
as  the  decimal  of  a  mile. 


12 

10.8    in. 

3 

9.9    ft. 

5i 

3.3    yds. 

320 

233.6    rds. 

0.73  mi.  Ans. 

3.   Express  71  sq.  rds.  54  sq.  ft. 

64.8  sq.  in.  as  the  decimal  of  an 

acre. 

144 

64.8      sq.  in. 

9 

54.45    sq.ft. 

30| 

6.05    sq.  yds. 

160 

71.20    sq.  rds. 

0.445  A.  Ans. 

4.   Express  15hrs.  14 min.  6 sec. 
i  the  decimal  of  2  days. 


6.000    sec. 
14.100    min. 
15.235    hrs. 


0.6348 
2 


0.6348  dys. 
=  0.3174.  Ans. 


5.  Express  38  sq.  rds.  21  sq. 
yds.  5  sq.  ft.  108  sq.  in.  as  the 
decimal  of  an  acre. 


144 

108.000  sq.  in. 

9 

5.750  sq.  ft. 

30i 

21.638  sq.  yds. 

160 

38.715  sq.  rds. 

0.242  A.  Am 

208 


ARITHMETIC. 


6.  Express  3  mi.  242  rds.  2  yds. 
2  ft.  3  in.  as  the  decimal  of  7  mi. 
160  rds. 


12 
3 

320 


3.00  in. 
2.25  ft. 
2.75  yds. 
242.5    rds. 


3201160 
7.5  mi. 


3.7578  mi. 


3.7578 
7.5 


0.501.  Am. 


7.   Express  5  hrs,  13  min.  30 
sec.  as  the  decimal  of  a  week. 


30.000    sec. 
13.500    min. 

5.225    hrs. 

0.2177  dy. 


0.0311  wk.  Ans. 


8.   Express  27°  14^  45'^  as  the 
decimal  of  90°. 


45.00^/ 
14.75^ 


27.246° 


27.246 
90 


=  0.303.  Am. 


9.   Express  54  dys.  2  hrs.  40 
min.  as  the  decimal  of  365 J  dys. 


54.1 
365^ 


40.0  min. 
2.6  hrs. 

54.1  dys. 

=  0.i48.  Am, 


10.  Express  2  lbs.  avoirdupois 
as  the  decimal  of  10  lbs  troy. 

'  2  lbs.  av.     =  14,000  grs.  troy. 
10  lbs.  troy  =  57,600  grs.  troy. 

14000 


57600 


0.243.  .4715. 


11.  Express  44,920.9025  hrs. 
as  the  decimal  of  a  year. 

1  yr.  =  8760  hrs. 


44920.9025 
8760 


=  5.128.  Am. 


12.  Express  1  drm.  avoirdupois  as  the  decimal  of  1  dwt.  troy. 
1  drm.  avoird.  =  ^1^  of  7000  troy  grs. 
=  27.344  troy  grs. 
1  dwt.  =  24  troy  grs. 
27.344 


24 


1.139.  Am. 


13.  Express  10  milligrams  as  the  decimal  of  a  grain,  if  a  kilogram 
equals  2  lbs.  8  oz.  3  dwt.  1  gr. 

2  lbs.  8  oz.  3  dwt.  1  gr.  =  15,433  grs. 
1  kg.  =  100,000  X  10  mg. 
15433 


100000 


-  0.15433.  Am. 


TEACHERS     EDITION. 


209 


14.   Express  14.52  sq.  yds.  as 
the  decimal  of  a  square  chain. 

1  sq.  eh.  =  16  sq.  rds. 
=  484  sq.  yds. 


14.52 

484 


0.03.  Ans. 


15.   Express  8  cwt.  77  1' 
oz.  as  the  decimal  of  a  ton. 


16 

100 

20 


9.600    oz. 
77.600    lbs. 
8.776    cwt. 


0.4388  t.  Ans. 


9.6 


Exercise  LIV. 

1.  Find  the  dilBference  in  longitude  between   two  places,  if  the 
difiference  in  time  be  1  hr.  15  min. 

1  hr.  15  min.  -  75  min.  =  ^  (75°)  =  18°  45^  Ans. 

2.  Find  the  difference  in  longitude  between   two  places,  if  the 
difference  in  time  be  2  hrs.  11  min. 

2  hrs.  11  min.  -  131  min.  =  ^  (131°)  =  32°  45^  Ans. 

3.  Find  the  difference  in  longitude   between   two   places,  if  the 
difference  in  time  be  5  hrs.  10  min.  10  sec. 

5  hrs.  10  min.  10  sec.  =  310  min.  10  sec.  =  ^  (310°  10^ 

=  77°  32^  30^^.  Ans. 

4.  Find  the  difference  in  longitude  between   two   places,  if  the 
difference  in  time  be  3  hrs.  25  min.  35  sec. 

3  hrs.  25  min.  35  sec  =  205  min.  35  sec.  =  |  (205°  35^) 

=  51°  23^  45^^.  Ans. 

5.  Find  the  difference  in  longitude  between  two  places,  if  the 
difference  in  time  be  6  hrs.  12  min.  30  sec. 


6  hrs.  12  min.  30  sec.  =  372  min.  30  sec. 


(372°  300 


93= 


30^^  Ans. 


6.  Find  the  difference  in  longitude  between   two   places,  if  the 
difference  in  time  be  4  hrs.  8  min.  12  sec. 

4  hrs.  8  min.  12  sec.  =  248  min.  12  sec.  =  ^  (248°  12^  =  62°  3^  Ans. 

7.  Find  the  difference  in  longitude  between   two   places,  if  the 
difference  in  time  be  18  hrs.  10  min. 

18  hrs.  10  min.  =  1090  min.  =  ^  (1090°)  =  272°  30^  Ans. 


210  ARITHMETIC. 


8.  Find  the  difference   in   longitude   between   two  places,  if  the 
difference  in  time  be  15  hrs.  15  min.  15  sec. 

15  hrs.  15  min.  15  sec.  =  915  rain.  15  sec.  =  |  (915°  15^ 

=  228°  48^  45'^  Ans. 

9.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  9°  20''. 

9°  20^  =  4  X  (9  min.  20  sec.)  =  37  min.  20  sec.  Ans. 

10.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  70°  30''. 

70°  30^  =  4  X  (70  min.  30  sec.)  =  4  hrs.  42  min.  Ans. 

11.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  56°  36^  12^^. 

56°  36'  12^'  =  56°  36.2'  =  4  x  (56  min.  36.2  sec.) 

=  3  hrs.  46  min.  24.8  sec.  Ans. 

12.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  108°  32'  36". 

108°  32'  36"  =  108°  32.6'  =  4  x  (108  min.  32.6  sec.) 

=  7  hrs.  14  min.  10.4.  sec.  Ans. 

13.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  120°  14'  30". 

120°  14'  30"  =  120°  14.5'  =  4  x  (120  min.  14.5  sec.) 
=  8  hrs.  58  sec.  Ans. 

14.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  100°  45'  54". 

100°  45'  54"  =  100°  45.9'  =  4  x  (100  min.  45.9  sec.) 

=  6  hrs.  4^  min.  3.6  sec.  Ans. 
'i 

15.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  2°  2'  2". 

2°  2'  2"  =  2°  23V^  =  4  X  (2  min.  2^  sec.)  =  8  min.  8^^  sec.  Am. 

16.  Find  the  difference  in  time  between  two  places,  if  the  differ- 
ence in  longitude  be  75°  10'. 

75°  10'  =.  4  X  (75  min.  10  sec.)  =  5  hrs.  40  sec.  Ans. 


teachers'  edition.  211 


Exercise  LV. 

The  longitude  of  some  public  building  in  : 

(1)  Berlin  is  13°  23^  43^^  E.  (7)  Jerusalem,  35°  32^  E. 

(2)  Rome,  12°  2V  W^  E.  (8)  Bombay,  72°  54^  E. 

(3)  Constantinople,  28°  59^  E,  (9)  Calcutta,  88°  19^  2^^  E. 

(4)  Pekin,  116°  23^  W  E.  (10)  Chicago,  87°  35^  W. 

(5)  SanFrancisco,122°26a5^'W.  (11)  New  York,  74°  0^  V^  W. 

(6)  St.  Louis,  90°  15^  15^^  W.  (12)  Montreal,  73°  25^  W. 

1.   When  it  is  noon  at  Greenwich,  what  is  the  clock-time  at  each 
of  the  above  places  ? 


(1) 

(5) 

13°  23^  43^^ 

122°  26^  lb'' 

=  13°23|r 

=  122°  26^^ 

=  4x(13min.  23f|8ec.) 

=  4  X  (122  min.  26^  sec.) 

=  53  min.  34if  sec.  p.m. 

=  8  hrs.  9  min.  45  sec. 

=  12  hrs.  53  min.  34||  sec. 

P.M. 

12  hrs. 

Ans. 

8  hrs.  9  min.  45  sec. 

(2) 

3  hrs.  50  min.  15  sec.  a.m. 

Ans. 

12°  27^  14^^ 

(6) 
90°  15^  15^^ 

=  12°  27^^ 

=  4  X  (12  min.  27^  sec.) 

=  90°  IbY 

=  49  min.  48|f  sec.  p.m. 

=  4  X  (90  min.  lb\  sec. 

) 

=  12  hrs.  49  min.  48^f  sec. 

P.M. 

=  6  hrs.  1  min.  1  sec. 

Ans. 

12  hrs.  M. 

(3) 

6  hrs.  1  min.  1  sec. 

28°  59^ 

6  hrs.  58  min.  59  sec.  a.m. 

Ans. 

=  4  X  (28  min.  59  sec.) 

(7) 
35°  32^ 

=  1  hi.  55  min.  56  sec.  p.m. 

Ans. 

=  4  X  (35  min.  32  sec.) 

(4) 

=  2  hrs.  22  min.  8  sec.  P.M. 

Ans. 

116°  23^  45^^ 

(8) 
72°  54^ 

=  116°  23f  ^ 

=  4  X  (116  min.  23f  sec.) 

=  4  X  (72  min.  54  sec.) 

=  7  hrs.  45  min.  35  sec.  p.m 

.Ans. 

=  4  hrs.  51  min.  36  sec.  p.m. 

Ans. 

212 


ARITHMETIC. 


(9) 
88°  19'  1" 
=  88°  19^' 

=  4  X  (88  min.  19^  sec.) 
»  5  hrs.  53  min.  IGj^  sec.  Am. 


(10) 

87°  35' 
=  4  X  (87  min.  35  sec.) 
=  5  hrs.  50  min.  20  sec. 

12  hrs.  M. 

5  hrs.  50  min.  20  sec. 

6  hrs.  9  min.  40  sec.  a.m.  Am. 


(11) 
74°  0'  V 
=  74°^' 

=  4  X  (74  mm.  -^  sec.) 
=  4  hrs.  56  min.  \  sec. 
12  hrs.  M. 

4  hrs.  56  min.  \  sec. 

7  hrs.  3  min.  59f  86c.  A.M.  Am. 

(12) 
73°  25' 

=  4  X  (73  min.  25  sec.) 
=  4  hrs.  53  min.  40  sec. 
12  hrs.  M. 

4  hrs.  53  min.  40  sec. 

7  hrs.  6  min.  20  sec.  a.m.  Ath. 


2.  When  it  is  half-past  four  p.m.  at  Chicago,  what  is  the  clock- 
time  at  each  of  the  above  places  ? 


•      (1) 

87°  35'  W. 
13°  23'  43"  E. 
100°  58'  43" 

- 100°  58ff' 
=  4  X  (100  min.  58f  ^  sec. 
=  6  hrs.  43  min.  54}f  sec. 
4  hrs.  30  min.  p.m. 

11  hrs.  13  min.  54||  sec.  p.m. 

.4.718. 


(2) 

87°  35'  W. 
12°  27'  14"  E. 
100°  2'  14" 

- 100°  2^' 

=  4x(100min.  2^sec.) 

—  6  hrs.  40  min.  8j^  sec. 

4  hrs.  30  min.  p.m. 

11  hrs.  10  min.  8j|  sec.  p.m. 

Afn, 


(3) 
87°  35'  W. 
28°  59'  E. 
116°  34' 

-  4  X  (116  min.  34  sec.) 
=  7  hrs.  46  min.  16  sec. 
4  hrs.  80  min.  p.m. 
12  hrs.  16  min.  16  sec.  A.M.  Ati^. 


(4) 
87°  35'  W. 
116°  23'  45"  E. 
203°  58'  45" 

360° 

203°  58'  45" 

156°  1'15" 


-156° 


156°  U' 

4x(l56  min.  1^  sec.) 
10  hrs.  24  min.  5  sec. 

4  hra.  30  min.  p.m. 
10  hrs.  24  min.  5  sec. 

6  hrs.  5  min.  55  sec.  A.M. 


Am. 


teachers'  edition.  213 


(5) 

122°  26i'  W. 

87°  35'  W. 

(9) 
88°  19'  2"  E. 
87°  35'        W. 

34°  51^' 
=  4  X  (34  min.  51^  sec.) 
=  2  hrs.  19  min.  25  sec. 

4  hrs.  30  min.  p.m. 

2  hrs.  19  min.  25  sec. 

2  hrs.  10  min.  35  sec.  p.m.  Ans. 

(6) 
90°  15^'  W. 
87°  35'  W. 

■      175°  54'  2" 

=  175°  543V. 

=     4  X  (175  min.  54^\y  sec.) 

=    11  hrs.  43  min.  363-2^  sec. 

4  hrs.  30  min.  p.m. 

16  hrs.  13  min.  363-2^  sec. 

=  4  hrs.  13  min.  S&j^^  sec.  a.m. 
Ans. 

2°  40^' 

=  4  X  (2  min.  40^  sec.) 
=  10  min.  41  sec. 

(10) 
(4  hrs.  30  min.  p.m.).  Ans. 

4  hrs.  30  min.  p.m. 

10  min.  41  sec. 

(11) 
87°  35'  W. 

4  hrs.  19  min.  19  sec.  p.m.  Ans. 

(7) 
87°  35'  W. 
35°  32'  E. 

74°    0'    3" 
13°  34'  57" 

123°    7' 
=   4  X  (123  min.  7  sec.) 
=    8  hrs.  12  min.  28  sec. 
4  hrs.  30  min.  p.m. 

=  13°  341§' 

=  4  X  (13  min.  34|^  sec.) 
=  54  min.  19f  sec. 
4  hrs.  30  min.  p.m. 

12  hrs.  42  min.  28  sec.  a.m.  5  hrs.  24  min.  19f  sec.  p.m.  Ans. 

Ans. 

(8) 
72°  54'  E. 
87°  35'  W. 


160°  29' 
=   4  X  (160  min.  29  sec.) 
=  10  hrs.  41  min.  §6  sec. 
4  hrs.  30  min.  p.m. 
15  hrs.  11  min.  56  sec. 
=    3  hrs.  11  min.  56  sec.  a.m. 

Ans. 


(12) 

87' 

>  35'  W. 

73' 

'  25'  W. 

14' 

»10' 

=  4x(14 

min.  10 

sec 

•) 

:56 

min. 

40  sec. 

4  hrs. 

30 

min. 

P.M. 

5  hrs. 

26 

min. 

40  sec.  ] 

P.M. 

Ans. 

214  ARITHMETIC. 


3.  When  it  is  eight  o'clock  a.m.  at  Constantinople,  what  is  th* 
clock-time  at  each  of  the  above  places  ? 

(1)  (5) 

28°  59'  E.  •  28°  59'  E. 

13°_23M3^E.  122°  26 J' W. 

15°  35'  17"  151°  25^' 

=  15°  35^'  _  4  ^  (151  jnin.  25^  sec.) 

=  4  X  (15  ram.  35^^  sec.)  ^  ^q  ^^^  5  ^^^^  4^  ^^^^ 


=  1  hr.  2  rain.  21f^  sec. 

8  hrs.  A.M 

1  hr.  2  min.  21t^  sec. 


8  hrs.  A.M. 
10  hrs.    5  min.  41  sec. 


^  ,       p^     .     _„-  „  .  9  hrs.  54  min.  19  sec.  p.m 

6  hrs.  57  mm.  38|f  sec.  a.m.  Am. 


Atik 


(2)  (6) 

28°  59'  E. 


12°  27'  14"  E. 
16°  31'  46" 


28°  59'  E. 
90°  15'  15"  W. 


119°  14'  15'-' 
=  16°31§§'  =nqoi4i/ 

=   4x(16min.  31|f  sec.)  .      ,iin  -..1 

,,  ^^     .     ^ /*^       '  =  4  X  (119  mm.  14i  sec.) 

=    1  hr.  6  mm.  7^^  sec.  m -,       tzn     •     c^ 

^  ^  =7  hrs.  56  mm.  57  sec. 

8  hrs.  A.M. 

Ihr.     6  min.    7,1^  sec.  8  hrs.  a.m. 

7  hrs.  56  mm.  57  sec. 

6  hrs.  53  min.  52}|  sec.  a.m.  Ans.  ] 

3  min.    3  sec.  a.m. 


(3) 
(8  hrs.  A.M.).  Am. 


=  12  hrs.  3  min.  3  sec.  a.m.  Ans. 


(7) 

(4)  35°  32'  E. 

116°  23f '  E.  28°  59'  E. 

28°  59'  E.  

6°  33' 

^'^°^^V  -4  X  (6  min.  33  sec.) 

-  4  X  (87  min.  24|  sec.)  „       26  min.  12  sec. 
«=  5  hrs.  49  min.  39  sec.  3  i^^s.  a.m. 

8  hrs.  A.M. 


8  hrs.  26  min.  12  sec.  a.m. 


1  hr.  49  min.  39  sec.  p.m.  Am.  Aru. 


teachers'  edition.  215 


(8) 
72°  5V  E. 
28°  59^  E. 

(10) 
87°  35^  W. 
28°  59^  E. 

43°  55^ 

116°  34^ 

=  4  X  (43  min.  55  sec.) 
=  2  hrs.  55  min.  40  sec. 

=  4  X  (116  min.  34  sec.) 
=  7  hrs.  46  min.  16  sec. 

8  hrs.  A.M. 

8  hrs.  A.M. 

10  hrs.  55  min.  40 

sec. 

A.M. 

7  hrs.  46  min.  16  sec. 

Arts. 

13  min,  44  sec.  a.m. 
=  12  hrs.  13  min.  44  sec.  a.m. 
Ans. 

(9) 
88°  19{  2^^  E. 

(11) 
74°  O^S^^W. 

28°  59^  E. 

28°  59^  E. 

59°  20^  2'-' 

102°  59^  3^^ 

59°20^V  =102°59^V 

4  X  (59  min.  20^V  sec.)         =  4  X  (102  min.  59^V  sec.) 
3  hrs.  57  min.  20^2^  sec.         =  6  hrs.  51  min.  56^  sec. 

8  hrs.  A.M.  8  hrs.  a.m. 

llhrs.57min.20Asec.A.M.  6  hrs.  51  min.  56^  sec. 

Ans.  1  hr.  8  min.  3f  sec.  a.m.  Ans. 

(12) 
73°  25^  W. 

28°  59^  E. 


102°  24' 
=  4  X  (102  min.  24  sec.) 
=  6  hrs.  49  min.  36  sec. 

8  hrs.  A.M. 

6  hrs.  49  min.  36  sec. 

1  hr.  10  min.  24  sec.  a.m.  Ans, 


216  ARITHMETIC. 


Exercise  LVI. 

When  it  is  noon  at  Greenwich  the  time  at 

(1)  Boston,  Mass.,  is  7  hrs.  15  min.  46  sec.  a.m. 

(2)  Augusta,  Me.,  7  hrs.  20  min.  40  sec.  a.m. 

(3)  Columbia,  S.C.,  6  hrs.  35  min.  32  sec.  a.m. 

(4)  Little  Rock,  Ark.,  5  hrs.  51  min.  12  sec.  a.m. 

(5)  Salt  Lake,  4  hrs.  30  min.  a.m. 

(6)  Albany,  N.Y.,  7  hrs.  5  min.  1  sec.  a.m. 

(7)  Columbus,  0.,  6  hrs.  27  min.  48  sec.  a.m. 

(8)  Harrisburg,  Penn.,  6  hrs.  52  min.  40  sec.  a.m. 

(9)  New  Orleans,  La.,  6  hrs.  a.m. 

(10)  Springfield,  111.,  6  hrs.  1  min.  48  sec.  A.M. 

(11)  Washington,  D.C.,  6  hrs.  51  min.  44  sec.  A.M. 

1.  What  is  the  longitude  of  each  of  the  above  places  ? 
(1)  (4) 

hra.  min.  seo.  hn.  min.  seo. 

12  0  0  12  0  0 

7  5  46  5  51  12 


4     44     14 

6     8     48 

=  284  min.  14  sec. 

=  J  of  284°  14' 

-  71°  3'  30^'  W.  Am. 

=  368  min.  48  sec. 
=  i  of  368°  48' 
=  92°  12'  W.  Am. 

(2) 

hn.      min.      mo. 

12     0     0 

7     20     40 

(6) 

hn.        min. 

12         0 

4        30 

4     39     20 

7      30 

=  279  min.  20  sec. 
=  \  of  279°  20' 
-  69°  50'  W.  Am 

=  450  min. 
=  iof450° 
=  112°  30'  W.  Am. 

(3) 

hri.      min.      aeo. 

12      0      0 

6     35     32 

(6) 

hn.      min.      aeo. 

12     0     0 
7     5     1 

5  24        .  28  4  54  59 

-  324  rain.  28  sec.  =  294  min.  59  sec. 

-  1  of  324°  28'  .=  ;^  of  294°  59' 

-  81°  7'  W.  Am.  -  73°  44'  45"  W.  Ans. 


teachers'  edition.  217 


(7) 

(9) 

hrs.               min.               seo. 

hrs. 

12             0             0 

12 

6           27           48 

6 

5           32           12 

6 

=  332  min.  12  sec. 

=  360  min. 

=  i  of  332°  12^ 

=  iof360° 

=  83°  3^  W.  Ans. 

=  90  °  W.  Ans. 

(8) 

(10) 

hrs.               min.               seo. 

hrs.              min.              sec. 

12             0             0 

12             0             0 

6           52           40 

6             1           48 

5             7        ,    20 

5           58           12 

=  307  min.  20  sec. 

=  358  min.  12  sec. 

=  \  of  307°  20^ 

=  ^  of  358°  12^ 

=  76°  50^  W.  Ans. 

^(11) 

=  89°  33^  W.  Ans. 

hrs. 

min. 

sec. 

12 

0 

0 

6 

51 

44 

T 

8 

16 

= 

308  min.  16  sec. 

= 

i  of  308°  16^ 

= 

77°  4^  W.  Ans. 

Exercise  LVII. 

-     1.  Reduce  7  gals.  3  qts.  ] 

L  pt.  to  gallons  and  decimal  of  a  gallon. 

2  l.OpO  pt. 

4  3.500  qts. 

7.875  gal. 

Ans. 

2.  Reduce  £4.375  to  pounds,  shillings, 

,  and  pence. 

£4.375 

20 

7.58. 

12 

ed 

£4  7s.  6d  Am. 

218 


ARITHMETIC. 


3.  Reduce  7.6875  gals,  to  gallons,  quarts,  and  pints. 

7.6875  gal. 
4 


2.75  qts. 
2 


1.5 

7  gals.  2  qts.  1.5  pts.  Ana. 

4.   Reduce  to  pounds,  shillings,  and  pence  f  5.875 ;  $  7.38 ;  $  17.85 ; 
$21.75 ;  if  $4.85  be  equal  to  a  pound. 


(1) 

(3) 

£'i.2H 

£3|H 

485)587.5 

485)1785 

0.2H  =  i¥i- 

m=u- 

^^^  of  20  s.  =  4^8. 

ff  of208.  =  13f?«. 

If  of  12cf.  =  2^^rf. 

^  of  12  d.  =  1^d. 

£14s.  2|^d  Ans. 

i 

£3  13  8.  7ffd  Am. 
(4) 

(2) 

£4fM 

^iHf 

485)2175 

485)738 

m=n- 

JIf  of208.  =  10f^«. 

f|of20s.  =  9fj«. 

tf  ofl2d  =  5^^d 

ff  of  12c?.  =  8ffd 

£1108.5^^^  ^7i«. 

£4  9s.8|fd  ilrw. 

5.  How  many  square  yards  in 

6. 

If  2  qts.  of  linseed  oil  be 

3.7156  acres? 

mixed  with  ^  pt.  spirits  of  turpen- 

3.7156 A. 

tine. 

what  fraction  of  the  mix- 

-    Xl60 

ture 

is  turpentine?     How  much 

turpentine   in   one  pint  of   the 

594.496  sq.  rds. 

mixture  ? 

xso\ 

2  qts.  =  4  pts. 

17983.504  sq.  yds.  Am. 

4pts.  +  ipt.  =  4ipts. 

±  =  2x1-1.           (1) 

4i     9^?     9               ^^^ 

i  of  1  pt.  =  i  pt.            (2) 

TEACHERS     EDITION. 


219 


7.    Reduce  5.1732  mi.  to  yards, 

feet,  and  inches. 

5.1732  mi. 

X  1760 

9104.832  yds. 

X3 

2.496  ft. 

Xl2 

5.952  iu. 

9104  yds.  2  ft.  5.952  in 

.  An?,. 

8.   If  a  man  walk  88  mi.  in 

26  hrs.,  how  many  feet  does  he 

walk  each  second  ? 

22       44 

»^x^^x    1    xi- 

9^8  ft. 

1  ''  1 ''  nn  %^ 

195 

15       13 

=  4Hf 

ft.  Am. 

9.  Of  a  mixture  of  sand  and 
lime  0.27  of  the  weight  is  lime. 
How  many  ounces  of  lime  in  a 
pound  of  the  mixture?  How 
many  troy  grains  of  lime  in  an 
avoirdupois  pound  of  the  mix- 
ture? 

16  oz.  0.27 

X  0.27  X  7000 


4.32  oz.  Am.      1890  troy  grs. 

10.   A  gill  of  water  is  put  into 

a  quart  measure,  and  the  measure 

filled  with  milk.     What  part  of 

the  mixture  is  water  ? 

8  gi.  =  1  qt. 

.-.  1  gi.  =  i  X  1  qt.  =  \  qt. 

.•.  \  is  water. 


11.   Reduce  555  ft.  to  the  deci- 
mal of  a  mile. 

0.1051136  mi.  Am. 
528)55.5000000 


12.   Reduce  1  mi.  13  rds.  2  yds. 
2  ft.  6  in.  to  inches. 


1  mi. 

X320 

320 

13 

333  rds. 

X51 

18331  yds. 

X3 

5502|  ft. 

Xl2 

66036  in.  Am 

13.  How  many  cubic  inches 
in  1\  cubic  feet  ? 

1728  cu.  in. 
X2| 

4320  cu.  in.  Ans. 

14.  How  many  pounds  avoir- 
dupois does  a  cubic  yard  of  water 
weigh  if  a  cubic  foot  weigh  1000 
ounces  ? 

27 
X  1000  oz. 

16)  27000  oz. 

1687i  lbs.  Am. 


220 


ARITHMETIC. 


15.  Express  the  weight  of  a 
cubic  yard  of  water  as  the  deci- 
mal of  a  ton. 

1687}  =,  675  _  21 
SOOOO         800  ~"  32' 

0.84375  t.  Ans. 
32)27.00000 


16.   What    is    the   weight 
7  bu.  3^  pks.  of  poL^toes? 

3^  pks.  =  M  bu.  =  I  bu. 

60  lbs. 
x7| 

472^  lbs.  ^rw. 


of 


17.  A  farmer  sowed  5  bu.  1  pk.  1  qt.  of  seed,  and  harvested  from 
it  103  bu.  3  pks.  5  qts.  How  much  did  he  raise  from  a  bushel  of 
seed? 


5  bu.  1  pk.  1  qt 
1  qt.  =  i  pk. 

lipk.  =  ^bu.  =  /^ 
5/,,  bu. 


:  103  bu.  3  pks.  5  qts. : 

5  qts.  =  f  pk. 

bu.  3f  pks.  =  ^  bu.  =  f  f  bu. 

103f  I  bu. 

5/j        169  ^^^ 

lit  of  4  pks.  =  2H|  pks. 
HI  of  8  qts.  =5^VVqts.      ' 
19  bu.  2  pks.  5.6  qts.  Ans. 


18.  How  many  bushels  in  5  t. 
of  oats? 

2000  lbs.    • 

5 

10000  lbs. 

312ibu.  ^718. 
32)10000 

19.  How  many  bottles,  each 
holding  1  pt.  3  gi.,  can  be  filled 
from  a  barrel  of  cider  ? 

1  pt.  3  gi. : 
3gi.-fpt. 

Hpt.-^gal.="Agal. 


8 
16      9 


144.  Am. 


20.  If  a  steamer  make  13  mi. 
6  rds.  an  hour,  how  far  will  she 
go  between  6  a.m.  and  6  p.m.? 
How  many  hours  will  she  re- 
quire to  make  113  miles? 


mi. 

13 


rd». 
6 

12 


156 


72.  Ans. 


13  mi.  6  rds. 


6  rds. 


113 


if^mi. 
^hh  mi. 
_  160      113 

2083        1 

18080 


2083 


Smilira 


TEACHERS     EDITION. 


221 


21.  If  a  locomotive  run  at  the 
rate  of  111  rds.  a  minute,  how 
many  hours  will  it  require  to 
run  from  Boston  to  Buffalo,  498 

miles  ? 

498  mi. 
X320 


159360  rds. 
1435ff 
111)159360 
6)143.5|f 

24  nearly.  Ans. 

22.  What  is  the  cost  of  12  A. 
146  sq.  rds.  land  at  1 16.25  an 

acre? 

146  sq.  rds.  =  |f  A. 

13 

12Hx^l6i=l^X^ 

16 

_|13429_  1 209.83.  Ans. 
64         * 

23.  What  is  the  cost  of  8  t. 

3  cwt.  27  lbs.  of  coal  at  |5.75  a 

ton? 

100127  lbs. 

20 1   3.27  cwt. 
8.1635  t. 
X$5| 


146.94.  Ans. 

24.   What  is  the  cost  of  7  t. 
1560  lbs.  of  hay  at  1 15.50  a  ton  ? 
1560  lbs.  =  iM^t.  =  xVVt. 
7.78 
X.$15^ 

$120.59.  Ans. 


25.  What  is  the  cost  of  a  car- 
load of  wheat  weighing  20,000 
lbs.,  at  $1.05  a  bushel? 

6)2000 
3331 
X$1.05 


1 350.  Ans. 


26.   Reduce  5  rds.  4  yds.  2|  ft. 
to  the  decimal  of  a  mile. 

ft. 
yds. 
rds. 
0.0184  mi.  A71S. 


27.    Reduce  9  sq.  ch.  11.25  sq. 
rds.  to  the  decimal  of  an  acre. 


3 

2.5 

^ 

4.83 

320 

5.87 

11.25 

.  9.703125 


0.9703125  A.    Ans. 


28.  Reduce    0.09375    bu.    to 
quarts. 

0.09375  bu. 
X32 

3  qts.  Ans. 

29.  Reduce    7560    chains    to 
miles. 

7560  ch. 
4 


30240  rds. 

94.5  mi.  Ans. 


32)3024.0 


222 


ARITHMETIC. 


\.   How  many  gross  are  2000 


pens? 


13|.  Am. 


144)2000 


31.   Find  the  cost  of  27.248  A. 
at  193.75  an  acre. 

27.248 
X$93| 


$2554.50.  Am. 

32.  Which  is  the  greater,  2.8 
of  3  ft.  11  in.  or  3.11  of  2  ft.  8  in., 
and  by  how  much  ? 

3  ft.  11  in.  2  ft.  8  in. 


Xl2 

47  in. 

X2.8 

131.6  in. 
99.52 


Xl2 

32  in. 
X3.ll 


99.52  in. 


12)32.08  in. 

2  ft.  8.08  in. 
The  former  by  2  ft.  8.08  in.  Am. 


33.  Reduce  171  lbs.  6  oz.  troy 
to  the  decimal  of  a  ton  avoir- 
dupois. 

7)0.1715 
0.0245 
X5760 


100 
20 


141.12  lbs.  avoird. 

141.12 
1.4112 


0.07056  t.  Am. 


34.    Express  14.52  sq.  yds.  as 
the  decimal  of  a  square  chain. 

30^)14.52 
4_ 

121)58.08(0.48  sq.  rd. 

0.03  sq.  ch.  Am 

16)0.48 


35.  If  a  sovereign  be  equal  to 
25.22  francs,  or  to  $4.85,  what 
decimal  of  a  dollar  is  a  franc? 

$0,192.  Am. 
2522)485.000 

36.  Express  2.805  florins  — 
1.89  half-crowns  as  the  decimal 
of  £0.472. 

2.805  1.89 

28.  X2.0  8. 

5.618.  4.725 «. 

4.725 

20)08858. 
0.04425 

0.09375.  Am. 


472)44. 1'oOOO 

37.  If  0.327  of  some  work  be 
done  in  3  hrs.  38  min.,  how  lon^ 
will  the  whole  work  require? 

3  hrs.  38  min. 

60 

218  min. 

666.6 

327)218000.0 

0.6  =  f  =  *. 
666.6  min.  =  666f  min. 
=  11  hrs.  6|  min. 
=-^11  hrs.  6  min.  40  sec.  Am. 


TEACHERS     EDITION. 


223 


38.  A  can  ran  a  mile  in  7.68 
min. ;  B  can  run  at  the  rate  of 
7.68  mi.  an  hour.  Which  is  the 
faster  runner  ? 

7.81 


768)6000.00 
.•.  A  is  the  faster  runner. 

39.  How  many  miles  an  hour 
does  a  person  walk  who  takes 
2  steps  a  second  and  1900  steps 
in  a  mile? 

60 
_><2 
120 
X60 

7200  steps. 
3|-f  mi.  Ans. 
19)72 

40.  If  an  ounce  troy  of  gold 
be  worth  $20,  what  is  the  value 
of  a  pound  avoirdupois  ? 

$20 

$240  per  lb.  troy. 
175  5 

W  Ans. 


41.  Two  stars  cross  the  merid- 
ian at  6  hrs.  4  min.  42.3  sec.  and 
7  hrs.  2  min.  57.21  sec,  respec- 
tively. What  is  the  interval  be- 
tween the  observations  ? 

hrs.  min.  sec. 

7  2         57.21 

6  4  42.3 


58 


14.91  A71S. 


42. 

How  long  will 

it  tak 

3    to 

fill   ^ 

of  a   cistern, 

when 

the 

whole 

requires  6  hrs.  10  min 
6  hrs.  10  min. 

? 

IC 

min.  =  U  lir.  = 
6i  hrs. 

=  ^-hr. 

6ixM-¥x 

M 

=  m  =  3^ 

hrs. 

tV?  of  60  min.  =  12f  min. 

I  of  60  sec.    =  24  sec. 
3  hrs.  12  min.  24  sec.  Ans. 

43.  The  circumference  of  a 
circle  is  6  yds.  1  ft.  5.1  in.,  and 
is  divided  into  360  degrees. 
What  is  the  length  of  55  degrees? 

6  yds.  1  ft.  5.1  in. 
X_3 
19  ft. 
X_12 
228 
X5.1 
233.1  in. 

3.2375 


72)233.1000 

3.2375  in. 
Xll 


12)35.6125  in. 

2  ft.  11.6125  in.  Ans. 

44.   Multiply  2  t.  16  cwt.  63| 
lbs.  by  If. 


t. 
2 

cwt. 

16 

lbs. 

63f 
X4 

)11 

6 

53f 

1 

2 

5 
16 

63| 

80tV.  Ans. 


224 


ARITHMETIC. 


45.   Into    how    many   shares 

46.   If  \\  of  one  line  be  equal 

has   £120  been    divided    when 

to  f  of  another  line,  which  is  the 

each  share  is  £3  8fi.  6fd.? 

greater  ? 

and  what  fraction  of  it 

is  the  less  ? 

6fd  =  ||«.  =  ^. 

13     8      39,  40 

15'    9         45 

8^.=£|  =  £f. 

.'.  the  former  is  the  greater.  Atii. 
3 

120      ^x^'^?     S5.Ans. 

f 

=  fx;^  =  |-- 

■^    U     1 

5 

47.   Multiply  5  mi.  206  rds.  2  ft.  2  in.  by 

r86. 

2  in.                             2  ft. 

206 

X  786                           X  786 

X786 

12)  1572  in.                       1572 

161916 

131  ft.                          131 

103 

3 

1703 

320)162019 

5i 

567  ..  2  ft. 

506..  99  rds. 

X2 

5                           11)1134 

X  786                                   103  ..  ^  yd 

3930 

606                   4436  mi.  99  rds.  \  yd 

.2  ft. 

4436  mi.          =  4436  m 

I.  99  rds.  1  yd 

.  6  in.  Am. 

48.  The  returns  of  a  gold  mine  are  241  t.  of  ore  yielding  2  oz. 
1  dwt.  15  grs.  of  fine  gold  a  ton,  and  193  t.  yielding  1  oz.  12  dwt. 
9  grs.  a  ton.     Find  the  value  of  the  whole  yield,  at  $19.45  an  ounca 


dwt. 

1 


15 
X241 


41 

9 

11 

15 

OI. 

dWU 

gr*. 

1 

12 

9 

nw. 

Xl93 

26 

0 

8 

9 

41 

9 

11 

15 

67 

X_12 

804 

10 

814  oz. 


10 


119.45 
X814 


$15,832.30  ilrw. 


teachers'  edition.  225 

49.    Divide  93  long  tons  56  lbs.  by  23  lbs.  5  oz. 

93  1.  t.  56  lbs.  23  lbs.  5  oz. 

X^20  X  16 

1860  1.  cwt.  368 

Xll2  5 

208320  37^  oz. 

56 


208376  lbs. 

Xl6  8938iff  Ans. 

3334016  oz.  373)3334016 

50.  Telegraph  poles  on  railroads  are  generally  erected  at  intervals 
of  88  yds.  Show  that  if  a  passenger  count  the  number  of  poles 
which  the  train  passes  in  three  minutes,  that  number  will  express 
the  number  of  miles  an  hour  the  train  is  going. 

1760  yds.  =  1  mi.  60  min.  =  1  hr. 

88  yds.  =  ^  mi.  3  min.  =  ^^  hr. 

51.  If  Greenwich  time  be  5  hrs.  8  min.  16  sec.  later  than  Wash- 
ington time,  and  Chicago  be  87°  35''  W.,  what  is  the  difference 
between  Washington  and- Chicago  time? 

87°  35^  =  4  X  (87  min.  35  sec.) 

=  5  hrs.  50  min.  20  sec. 

5  hrs.    8  min.  16  sec. 


42  min.    4  sec.  Ans. 

Exercise  LVIII. 

1.  A  train  from  New  York  to  Philadelphia,  90  miles,  makes  the 
whole  distance  in  2  hrs.  5  min.     What  is  its  rate  ? 

2  hrs.  5  min.  =  2^^  h^^- 

90  mi.  H-  2^2  =  43|  mi.  Ans. 

2.  Winlock,  in  1869,  found  that  electricity  went  through    7200 
miles  of  wire  in  f  of  a  second.     What  was  its  rate  per  second  ? 


226  ARITHMETIC. 


3.  If  the  time  required  for  a  signal  to  pass  through  the  cable  from 
Brest  to  Duxbury,  3799  miles,  be  0.816  of  a  second,  what  is  the  rate 
per  second  ? 

3799  mi.  ^  0.816  =  4655.637  mi.  Ans. 

4.  If  the  report  of  a  gun  1\  miles  distant  is  heard  in  5|  seconds 
after  the  flash  is  seen,  what  is  the  velocity  of  sound,  in  feet,  per 
second  ? 

1|  mi.  -J-  5f  =  f  mi.  -  ll73i  ft.  Ans. 

5.  If  a  man  walk  3|  miles  in  46  minutes,  what  is  his  rate  per 
hour? 

3|  mi.  -*-  f  ^  =  4/j  mi.  Ans. 

6.  If  a  horse  go  47^  miles  in  10  hrs.  40  min.,  what  is  his  average 
rate  per  hour  ? 

10  hrs.  40  min.  =  lOf  hrs. 
47^  mi.  -^  lOf  =  4f  I  mi.  Ans. 

7.  If  a  stone  on  a  glacier  move  95^  feet  in  188  days,  what  is  its 
rate,  in  inches,  per  day  ? 

951  ft.  ^  188  =  m  ft.  =  6^\  in.  Ans. 

8.  If  a  horse  trot  f  of  a  mile  in  2^  minutes,  in  what  time  can  he 
trot  a  mile  ' 

2|  min.  -5-  f  =  2f  min.  Ans. 

9.  If  a  train  run  18  miles  in  39  minutes,  how  long  does  it  take  to 
run  one  mile  ? 

39  min.  -4- 18  =  2^  min.  Ans. 

10.  If  sound  travel  1125  feet  a  second,  how  long  will  it  take  to 
travel  one  mile  ? 

1  mi.  =  5280  ft. 
5280 +  1125  =  4.7  sec.  Am. 

11.  If  a  train  require  3  hours  to  travel  104 J  miles,  find  its  aver- 
age time  for  travelling  a  mile. 

104}  mi.  +  3  =  34|  mi. 
60  min.  +  84|  =  \\^  min.  =  1  min.  43^  sec.  Ans. 


teachers'  edition.  ^  227 

12.  If  a  mower  cut  7|  acres  of  grass  in  3^  days,  what  part  of  a 
day  will  it  take  him  to  cut  one  acre  ?  If  a  day  consist  of  10  work- 
ing-hours, what  part  of  an  acre  does  he  cut  in  an  hour? 

3idys.-^7i  =  x'3dy.  Ans. 
SI  dys.  of  10  hrs.  =  35  hrs., 
and  7|  A.  ^  35  =  y3__  ^    ^,^5^ 

13.  If  a  mower  cut  3|  square  rods  in  ^  of  an  hour,  how  many 
acres  can  he  cut  in  a  day  of  10  hours? 

3|  sq.  rds.  -J- 1-  =    28  sq.  rds. 
10  X  28  sq.  rds.  =  280  sq.  rds.  =  If  A.  Ans. 

14.  If  a  fountain  yield  117|-  gallons  in  |  of  an  hour,  at  what  rate 
per  hour  is  it  flowing? 

117|  gals.  ^  f  =  156|  gals.  Ans. 

15.  If  a  merchant's  profits  be  $3147  in  7^  months,  what  are  his 
profits  for  a  year  ? 

7irao.=  j|yr.  =  |-yr. 
$3147^1  =  15035.20.  Ans. 

16.  If  a  wheel  turn  17°  30^  in  35  minutes,  in  how  many  hours 
does  it  make  a  complete  revolution  ? 

17°  30^  -  35  =  30^  =  1°  in  one  min. 
360  ^  I  =  720  min.  =  12  hrs.  Ans. 

17.  If  a  man's  expenditures  be  $4358  in  13^  months,  what  is  his 
yearly  rate  of  expenditure  ? 

13^ 

i3i  "^-  =  -^  yr.  =  -V-  yr. 

$4358-^1^  =  $3922.20.  Ans. 

18.  If  a  cistern  lose  by  leakage  7  gals.  1  pt,  in  49  hrs.  40  min., 
what  is  its  hourly  rate  of  loss  ? 

49  hrs.  40  min.  =  49f  hrs. 
7  gals.  1  pt.  =  57  pts. 
57  pts.  ^  49f  =  1^^\  pts. 


228  ARITHMETIC. 


19.  If  the  circumference  of  the  earth  at  the  equator  be  24,900 
miles,  at  what  rate  per  hour  is  a  person  there  carried  round,  one 
whole  rotation  being  made  in  23  hrs.  56  min.  ? 

23  hrs.  56  min.  =  23}f  hrs. 
24,900  mi.  -s-  23^  =  1040^f  ^  mi.  Am. 

20.  If  a  man  travel  ^  miles  m  7^  minutes,  how  many  miles  will 
he  travel  in  50  minutes  ?  and  how  long  will  he  take  to  travel  50 
miles  ? 

7\  min.  -4-  3f  =  2^^  min. 
50  mi.  -f-  2^  =  24  mi.  Ans. 
3f  mi.  ^  7^  =  H  rai. 
50  -5-  H  =  104^  min. 

=  1  hr.  44  min.  10  sec.  Ans. 

21.  If  A  can  mow  a  certain  meadow  in  4  days,  and  B  in  3  days, 
how  long  will  it  take  both  ? 

If  A  can  mow  it  in  4  days,  in  one  day  he  can  mow  ^  of  it. 
If  B  can  mow  it  in  3  days,  in  one  day  he  can  mow  -^  of  it. 
Both  together  can  mow  ^  +  ^  =  ^7^  of  it  in  one  day. 
.-.  both  together  can  mow  the  whole  in  -^^  days,  or  l^days.  Ans. 

22.  If  A  can  lay  a  certain  wall  in  4|  days,  and  B  in  5^  days,  how 
long  will  it  take  both  ? 

If  A  can  do  it  in  i^  days,  in  one  day  he  can  do  —  =  f  of  it. 

If  B  can  do  it  in  5J  days,  in  one  day  he  can  do  —  =  A-  of  it. 

Both  together  can  do  f  +  ^  =  f  §  of  it  in  one  day. 

.-.  both  together  can  do  the  whole  in  |^  days,  or  2^  days.  Ans. 

23.  If  a  pipe  will  fill  a  vessel  in  4^  hours,  and  another  in  3 J  hours, 
how  long  will  it  take  both  to  fill  the  vessel? 

If  one  pipe  will  fill  it  in  4J  hrs..  in  one  hr.  it  will  fill  —  —  f  of  it 

If  another  pipe  will  fill  it  in  3  J.hr8.,  in  one  hr.  it  will  fill  —  =  ^  of  it 

Both  pipeB  together  will  fill  f  +  f  -  ^f  of  it  in  one  hour. 
.-.  both  pipes  together  will  fill  it  in  f4=  Ifi  hrs.  =  1  hr.  58  min. 
7i  sec.  Ans. 


teachers'  edition.  229 

24.  If  A  can  go  from  Boston  to  Albany  in  9|  hours,  and  B  from 
Albany  to  Boston  in  11^  hours,  and  they  start  at  the  same  time,  in 
how  many  hours  will  they  meet? 

If  A  can  go  in  9|  hrs.,  in  1  hour  he  can  go  —  =  /y  of  the  distance. 

If  B  can  go  in  11|  hrs.,  in  1  hour  he  can  go  —  =  ^\  of  the  distance. 

Both  together  can  go  /y  +  ^\  =  j^^y^  of  the  distance  in  1  hour. 
.-.  the 

25.  A  requires  4  days,  B  3  days,  and  C  4|  days,  to  do  a  certain 
piece  of  work.     How  long  will  it  take  all  three  working  together  ? 

If  A  can  do  it  in  4  days,  in  one  day  he  can  do  ^  of  it. 
If  B  can  do  it  in  3  days,  in  one  day  he  can  do  ^  of  it. 

If  C  can  do  it  in  4|  days,  in  one  day  he  can  do  —  =  f  of  it. 

All  together  can  do  |  +  |-  +  |  =  ff  of  it  in  one  day. 

.•.  it  will  take  them,  all  working  together,  ||  days  =  IgV-  -4?2s. 

26.  A  can  mow  f  of  a  field  in  3  days  ;  B  can  mow  |  of  it  in  4  days. 
How  long  will  it  take  both  to  mow  the  field  ? 

3  days  -^  f  =  5|  days,  and  4  days  -4-  |  =  6  days. 

If  A  can  mow  it  in  5f  days,  in  one  day  he  can  mow  —  =  /y  of  it. 

If  B  can  mow  it  in  6  days,  in  one  day  he  can  mow  |  of  it. 
Both  together  can  mow  y\  +  i  =  if  of  it  in  one  day. 
.•.  both  together  can  niow  it  in  ff  days  =  2|f  days.  Ans. 

27.  One  pipe  can  fill  a  cistern  half  full  in  f  of  an  hour,  and 
another  can  fill  it  three-quarters  full  in  |  an  hour.  How  long  will 
it  take  both  pipes  to  fill  the  cistern  ? 

f  hr.  -r- 1  =  1|^  hrs.,  and  |  hr.  -=- 1-  =  f  hr. 

If  one  pipe  fills  it  in  1|  hrs.,  in  one  hour  it  will  fill  —  =  f  of  it. 

If  another  pipe  fills  it  in  f  hr.,  in  one  hour  it  will  fill  -  =  |  of  it. 


Both  together  will  fill  |  +  f  =  -y-  of  it  in  one  hour. 
.-.  both  together  will  fill  it  in  6  ^  13  =  j%  hr.  Ans. 


230  ARITHMETIC. 


28.  A  ci-stern  which  holds  100  gallons  can  be  filled  from  a  pipe  in 
25  minutes,  and  emptied  by  a  waste-pipe  in  45  minutes.  If  both  are 
opened  together,  how  long  will  it  take  to  fill  the  cistern,  and  how 
much  water  will  be  wasted  ? 

The  water-pipe  fills  ^  every  minute. 

The  waste-pipe  empties  -^  every  minute. 

When  both  are  open,  -^^  —  ;i^  =  ^f^  is  gained  every  minute. 

.•.  the  whole  will  be  filled  in  -^f^  =  56^  min.  Ans. 

If  -^^  of  the  cistern  is  wasted  every  minute,  in  56 J  minutes  66^  x  ^^ 

would  be  wasted. 
Now,  as  the  cistern  holds  100  gals.,  the  number  of  gallons  wasted 

would  be  56^  X  :^  X  100  gals. 

5  25 

bQ\  X  ^V  X  100  =  2?f  X  :^  X  ^?^  =  125  gals.  Ans. 


29.  A  pipe  can  fill  a  cistern  one-third  full  in  \  of  an  hour ;  a 
waste-pipe  can  empty  \  of  the  cistern  in  20  minutes.  If  both  pipes 
are  opened,  in  what  time  will  the  cistern  be  filled? 

^  hr.  X  3  =  f  hr.  =  45  min.,  and  20  min.  x  4  =  80  min. 

The  water-pipe  fills  -^-^  every  minute. 

The  waste-pipe  empties  -^j^  every  minute. 

When  both  are  open,  ^  —  ^i^  =  ^^^  is  gained  every  minute. 

.-.  the  whole  will  be  filled  in  ifa  =  102^  min. 

=»  1  hr.  42  min.  51f  sec.  Ans. 


80.  If  one  pipe  runs  into  a  cist6rn  at  the  rate  of  2  gallons  in  3 
minutes,  and  another  at  the  rate  of  5  gallons  in  4  minutes,  while  the 
water  is  running  out  of  a  third  pipe  at  the  rate  of  4  gallons  in  5 
minutes,  how  long  will  it  take  to  gain  71  gallons  in  the  cistern  ? 

2  gals.  -1-3  =  1  gals.,  5  gals,  -i-  4  =  J  gals.,  and  4  gals.  -«-  5  =  f  gala. 
If  one  pipe  pours  in  |  gals,  per  minute,  another  pours  in  \  gals,  per 

minute,  and  another  empties  |  gals,  per  minute,  the  cistern 

gains  I  +  J  —  I  =  1^  gals,  per  minute. 
.'.  it  will  take  as  many  minutes  to  gain  71  gals,  as  71  -»-  f^  =  63}f 

min.  =  1  hr.  3  min.  34f  f  sec. 


teachers'  edition.  231 

31.  A  and  B  can  do  a  piece  of  work  in  2i  days ;  A  and  C  in  Si- 
days ;  B  and  C  in  4|-  days.  Required  the  time  in  which  all  three, 
working  together,  can  do  the  work,  and  in  which  each  can  do  it 
alone. 

If  A  and  B  can  do  it  in  2i  days,  they  can  do  —  =  f  of  it  in  one  day. 
If  A  and  C  can  do  it  in  3|  days,  they  can  do  —  =  ^^  of  it  in  one  day. 

If  B  and  C  can  do  it  in  4|  days,  they  can  do  —  =  -^^  of  it  in  one  day. 

All  can  do  f  +  x%  +  tV  =  xft  ^"^  2  days,  or  \\%  of  it  in  one  day. 

.-.  all  can  do  it  in  \\%  =  2j2j2_  days.  An8. 

If  A,  B,  and  C  can  do  \\%,  and  B  and  C  j*y  of  it  in  one  day,  A  can 

do  \\%  -  tV  -  3T0  of  it  in  one  day. 
,'.  A  can  do  the  whole  in  ^f^-  =  4ff  days.  An%. 
If  A,  B,  and  C  can  do  \\%,  and  A  and  C  y%  of  it  in  one  day,  B  can 

do  \\%  -  A  =  sVo  of  it  in  one  day. 
.'.  B  can  do  the  whole  in  ^^  =  5f f  days.  Ans. 
If  A,  B,  and  C  can  do  ^|f ,  and  A  and  B  f  of  it  in  one  day,  C  can 

do  iH  - 1  =  3  ¥%  of  it  in  one  day. 
.'.  C  can  do  the  whole  in  -\*^  =  14|f  days.  Ans. 

32.  Sampson  &  Reed  sold  f  of  a  lot  of  wheat  to  one  party,  |  of 
the  remainder  to  another,  and  had  93  bushels  left.  How  much  had 
they  at  first  ? 

After  selling  f  of  the  wheat  they  had  f  left. 

After  selling  f  of  f  of  the  wheat  they  had  ^  of  f  =  -^^  left. 

Then  93  bush.  =  ^\  of  the  lot. 

.'.  the  whole  lot  =  93  bush.  -^  j\  =  992  bush.  Ans. 

33.  In  a  certain  school  -^^  of  the  scholars  are  girls,  f  of  the  boys 
are  over  IG  years  old,  and  6  boys  are  under  16.  How  many  girls, 
and  how  many  scholars  in  all  ? 

After  the  girls,  or  y\  of  the  school,  are  taken  out,  there  remain  y%  of 

the  school,  or  the  boys. 
After  |-  of  j^^  are  taken  out,  f  of  j^^  =  j\  are  left. 
.-.  the  whole  number  of  scholars  is  6  -^  y\  ^  32  scholars.  Ans. 
■^^  of  32  scholars  =  18,  number  of  girls.  Ans. 


232  ARITHMETIC. 


34.  In  a  certain  school  ||  are  boys ;  ^j  of  the  girls  are  under  16, 
and  13  girls  are  over  16.  How  many  boys  and  how  many  girls  in 
the  school  ? 

After  the  boys,  or  ^|  of  the  school,  are  taken  out,  there  remains 

^^  of  the  school,  or  the  girls. 
After  /j  of  H  are  taken  out,  |f  of  ^\  =  if  are  left. 
.-.  the  whole  number  of  scholars  =  13  -j-  ^f  =  48  scholars, 
ii  of  48  =  22  girls.  Ans.     ^f  of  48  =  26  boys.  Ans. 

35.  If  from  a  certain  number  |  of  it  be  subtracted,  then  ^  of  the 
remainder,  then  j-  of  that  remainder,  and  6  still  remain,  what  is  the 
number? 

After  f  of  it  is  subtracted,  ^  is  left. 
After  ^  of  I  is  subtracted,  |  of  |  =  |  is  left. 
After  }  of  I  is  subtracted,  f  of  |  =  s%  is  left. 
.-.  the  number  =  6  -s-  /^  =  35.  Ans. 

36.  20  is  f  of  f  of  f  of  what  number  ? 

f  of  ^  of  ^  =  -•        .-.  the  number  =  20  -^  4  =  120.  Ans. 
^^36  ^ 

2 

37.  6  is  f  of  ^  of  i  of  what  number  ? 

-  of  ^  of  -  =  — .       .-.  the  number  =  6  -^  A  =  35.  Ans. 
7      5      ^     35  ^ 

38.  Express  ^^  of  1  lb.  troy  +  ^  of  1  lb.  avoirdupois  as  troy  and 
as  avoirdupois  weights. 

175 


1 

29 

lb.  av.  = 

29   W^ 
144 

175 
4176 

lbs.  troy. 

1 

29 

+  175^ 
4176 

144  +  175 
4176 

319 
4176 

-1^- 

tVi  lbs.  =  tV*  of  12  oz.  =  ji  oz.  =  li  of  20  dwt.  =  18J  dwt 

=  18  dwt.  8  grs.  Ans. 

iilbs.troy-jyLof?^=lll,^^^ 

144  ^     IH      }fm     175 

175 

^Yy  lbs.  —  ^yy  of  16  oz.  =>  1  jfy  oz.  av.  Ans. 


teachers'  edition.  233 

39.  The  cargo  of  a  ship,  worth  1 45,000,  belongs  to  three  partners. 
A  owns  I  of  f  of  it,  B's  share  is  equal  to  3^^  of  f  of  A''s  share,  and 
C  owns  the  remainder.     What  ought  each  to  receive  from  the  sale  ? 

I  of  f  =  iV  of  ship.     tV  of  $45,000  =  $  21,000,  A's  share. 

3^3^  of  f  of  tV  =  i  of  ship,     i  of  $  45,000  =  $  1 5,000,  B's  share. 

$45,000  -  ($21,000  +  $15,000)  =  $9000,  C's  share. 

40.  Find  the  largest  number  which  is  contained  an  integral  num- 
ber of  times  in  each  of  the  following  :  2|,  6j^^,  11^,  19^. 

2f,6A,iH,i9i  =  -^/.W-.¥.H^. 

G.  CM.  of  23,  115,23,  115  =  23. 

L.  CM.  of  9,  18,  2,6  =  18. 

.-.  G.  C  M.  of  fractions  =  ff  =  lyV  ^^s- 

41.  A  person  bequeathed  j\  of  his  property  to  A,  ^  of  it  to  B,  | 
to  C,  I  to  D,  and  the  remainder,  $550,  to  E.  What  was  the  value 
of  the  whole  property  ? 

A  +  i  +  i  +  i  =  ff- 

After  If  is  subtracted,  y^^  remains. 

.-.  whole  money  =  $550  h-  ^\  =  $  13,200.  Ans. 

42.  Arrange  in  descending  order  of  magnitude,  |f ,  |f ,  |f . 

13     15     16  _  9126,  10125,  10400 

25'    26'    27  17550 

.•.  the  order  of  magnitude  is  |f,  |f ,  ^f .  Ans. 

43.  A  bankrupt's  debts  are  $  2520,  and  the  value  of  his  property 
is  $1890.     How  much  can  he  pay  on  a  dollar? 

2520        4       ^ 

44.  A  bankrupt's  debts  are  $4264,  and  he  pays  62^  cents  on  a 
dollar  ;  what  are  his  assets  ? 

533 

0.62|  =  -.  -  of  ^-^^  =  $  2665.  Ans. 

8  8  1 


234  ARITHMETIC. 


45.   If  15  yards  of  silk  cost  $  18.75,  how  much  will  20^  yards  cost  ? 


If  15  yds.  cost  $18|,  1  yd.  costs  ^^, 

15 

and  20J  yds.  cost  20^  x  ^^• 
15 


20}xii|i  =  ^xf  x^  =  i|25  =  |25.42.  An.. 
15        S      Xp        4  12 


46.   If  3f  pounds  of  tea  cost  f  3.80,  how  much  can  I  buy  for  1 21.87  ? 


If  3|  lbs.  cost  $3.80,  1   lb.  costs  ^^^,  and  as  much  can  be 


bought  for  121.87  as  2187  ^  ^• 
2187  ^'^  =  i^x^hX^^  =  -V^  =  19{m  lbs.  Am. 

47.   If  x\  of  a  ton  of  coal  cost  $  1.12,  what  is  the  price  of  5J  cwt.  ? 
5icwt.  =  |t..  =  -Ht. 
If  ^j  t.  cost  $  1.12,  1  t.  costs  ^^,  and  ^  t.  costs  ^  x  ?^^. 

11     |W     14_|21L56_. 

^^"^     1     "^3"     15     -*1-^-^^- 

5 


48.   If  fV  of  a  i)iece  of  work  be  done  in  25  days,  how  much  will 
be  done  in  Uf  days? 

If  ^  can  be  done  in  25  days,  the  whole  can  be  done  in  —  days, 

"A" 
and  as  much  can  be  done  in  11 1  days  as  11|  +  — • 

7  * 

118^25      2  ^  1  ^??      14      . 


teachers'  edition.  235 

49.  A  man  walks  18  mi.  106  rds.  3f  yards  in  5|  hours.     How 
long  does  he  take  to  walk  a  mile  and  a  half? 

18  mi.  106  rds.  3f  yds. :  ^^,  ^^  =  lOSI  _  J  mi. 

■^2  vd^  -  M  rd   -  ^  rd  ^^^ 

'  ^      ~  51       "  '  18  mi.  106  rds.  3f  yds.  =  181  mi. 

1 81 
If  he  walks  18|^  mi.  in  5^  hrs.,  he  will  walk  — ^  mi.  in  1  hr,, 

and  it  will  take  him  as  long  to  walk  l^  miles  as  1^  -. ^ 

li^lM  =  ilx  — X-  =  — hr.-27min.  Ans. 
'       5i       2  ^^     2     20 
o 

50.  When  an  ounce  of  gold  is  worth  $19.46,  what  is  the  value  of 
0.04  of  a  j)0und  ? 

$19.45  X  12  X  0.04  =  $9,336.  Ans. 


51.  If  9  horses  can  plow  46  acres  in  a  certain  time,  how  many 
acres  can  12  horses  plow  in  the  same  time  ? 

Since  9  horses  can  plow  46  acres  in  a  certain  time, 
1  horse  can  plow  ^  of  46  acres  in  the  same  time, 
and  12  horses  can  plow  12  X  ^  of  46  =  61^  acres.  Ans. 

52.  If  12  men  can  reap  a  field  in  4  days,  in  what  time  will  32  men 
reap  it? 

Since  12  men  can  reap  a  field  in  4  days, 
1  man  can  reap  it  in  12  X  4  days, 

and  32  men  can  reap  it  in  ■  ^         days  ==  1^  days.  Aiis. 

53.  If  72  men  dig  a  trench  in  63  days,  in  how  many  days  will 
42  men  dig  another  trench  three  times  as  great? 

Since  72  men  can  dig  a  trench  in  63  days, 
1  man  can  dig  it  in  72  X  63  days. 
1  man  can  dig  one  3  times  as  large  in  3  X  72  X  63  days, 

and  42  men  can  dig  it  in  ^  X  '^^  X  ^^  days  =  324  days.  Ans. 


23G  ARITHMETIC. 


54.  If  a  rnan  travels  540  miles  in  24  days,  walking  6  hours  a  day, 

how  many  miles  can  he  travel  in  3  days,  walking  8  hours  a  day  ? 

Since  he  can  go  540  mi.  in  24  X  6  hrs.  =  144  hrs., 
in  one  hour  he  can  go  y^  of  540  mi., 
and  in  3  X  8  hrs.  =  24  hrs.  he  can  go  24  X  xt¥  ^^  ^"^^  ™^- 

=  90  mi.  Atu. 

55.  If  15  men  can  perform  a  piece  of  work  in  22  days,  how  many 
men  will  finish  another  piece  of  work  four  times  as  large  in  ^  of 
the  time  ? 

Since  15  men  can  do  the  work  in  22  daySf 

it  will  take  4x15  men  to  do  4  times  the  work  in  22  days, 
and  to  do  4  times  the  work  in  ^  of  22  days  it  will  take 

5  X  4  X  15  men  =  300  men.  Ans. 

56.  A  garrison  of  2100  has  provisions  for  9  months,  but  receives 
reinforcements  of  600  men.     How  long  will  the  provisions  last? 

Since  the  provisions  will  last  2100  men  9  months, 
they  will  last  1  man  2100  X  9  months, 
and  they  will  last  2100  +  600  =  2700  men 

^1^><1  mos.  =  7  mos.  Ans. 
2700 

57.  If  a  cubic  foot  of  ice  weigh  57f  pounds,  how  many  cubic  feet 
of  ice  will  weigh  a  ton  ? 

Since  1  cubic  foot  of  ice  weighs  57|  lbs. ;  to  weigh  a  ton 

it  will  take  ^  cu.  ft.  =  34f|^  cu.  ft.  Ana. 
57f 

58.  How  many  bushels  of  wheat  will  serve  72  people  8  days 
when  4  bushels  serve  6  people  24  days? 

72  people  will  eat  twelve  times  as  much  as  6  people  in  the  same 

time. 
And  the  same  number  of  people  will  eat  |  as  much  in  8  days  as 

in  24  days. 
Hence,  72  people  in  8  days  will  eat  12  x  |  times  as  much  as  6 

people  in  24  days. 

12  X  i  X  4  bushels  =  16  bushels.  Ans. 


teachers'  edition.  237 

59.  If  2  horses  eat  8  bushels  of  oats  in  16  days,  how  many  horses 
will  eat  3000  bushels  in  24  days  ? 

In  16  days  8  bu.  can  be  eaten  by  2  horses. 

In  1  day  8  bu.  can  be  eaten  by  16  x  2  horses. 

16  X  '' 
In  1  day  1  bu.  can  be  eaten  by  — ^^^-^  horses. 

8 

1 6  V  ^ 
In  24  days  1  bu.  can  be  eaten  by horses. 

^  -^  24  X  8 

In  24  days  3000  bu.  can  be  eaten  by  ^^^Q  X  16  X  2  y^^^ 
^  -^         24x8 

=  500  horses.  Ans. 

60.  If  a  man  travel  150  miles  in  5  days,  when  the  days  are  12 
hours  long,  in  how  many  days  of  10  hours  each  Avill  he  travel  500 
miles  ? 

He  can  go  150  miles  in  5  days  of  12  hours  =  60  hours. 

He  can  go  1  mile  in  j%%  hour. 

TT  KAA      -1      •     500  X  60 1 

He  can  go  500  miles  m —  hours. 

loU 

He  can  so  500  miles  in  — —  days  of  10  hours, 

^  150x10      "^ 

=  20  days.  Aiis. 

61.  If  a  regiment  of  939  soldiers  consume  351  bushels  of  wheat  in 
21  days,  how  many  soldiers  will  consume  1404  bushels  in  7  days  ? 

1404  bu.  will  last  the  same  number  of  men  four  times  as  long 

as  351  bu. 
And  the  same  amount  will  last  three  times  the  number  of  men 

for  7  days  as  for  21  days. 

3  X  4  X  939  soldiers  =  11,268  soldiers.  Ans. 

62.  If  5  men  can  reap  a  field  of  12i  acres  in  3|-  days,  working 
16  hours  a  day,  in  what  time  can  7  men  reap  a  field  of  15  acres, 
working  12  hours  a  day  ? 

5  men  can  reap  12-^-  acres  in  3|-  days  of  16  hours  =  56  hours. 

1  man  can  reap  12^  acres  in  5  x  56  hours. 

1  1  •     5  X  56  T_ 

1  man  can  reap  1  acre  in  -^  —  hours. 

^  12i 


238  ARITHMETIC. 


15  X  5  X  56 
1  man  can  reap  15  acres  in  -^—^ — — —  hours. 

12^ 


15  X  5  X  56 
7  men  can  reap  15  acres  in  -^-^ — ^-^-^  hours. 
7x12^ 

15  )^  5  X  56 
12  X  7  X  12J 


15  ^^  5  X  56 
7  men  can  reap  15  acres  in days  of  12  hours, 


4  days.  Ans. 


63.  If  7  men  mow  22  acres  in  8  days,  working  11  hours  a  day, 
in  how  many  days,  working  10  hours  a  day,  will  11  men  mow 
360  acres  ? 

7  men  can  mow  22  acres  in  8  days  of  11  hours  =  88  hours, 

1  man  can  mow  22  acres  in  7  X  88  hours. 

7  X  88 
1  man  can  mow  1  acre  in  —^ —  hours. 
22 

7  X  88 

12  men  can  mow  1  acre  in  hours. 

12x22 

12  men  can  mow  360  acres  in —  hours. 

12x22 

12  men  can  mow  360  acres  in  ^^Q  X  7  X  88  ^        ^^  ^^  ^^ 
10  X  12  x  22      -^ 

=  84  days.  Ans. 

64.  If  44''cannon,  firing  30  rounds  an  hour  for  3  hours  a  day, 
consume  300  barrels  of  powder  in  5  days,  how  long  will  400  barrels 
last  Q6  cannon,  firing  40  rounds  an  hour  for  5  hours  a  day  ? 

44  cannon  firing  30  rounds  for  3  hours  consume 

300  bbls.  in  5  days. 
44  cannon  firing  30  rounds  for  1  hour   consume 

300  bbls.  in  3  X  5  days. 
44  cannon  firing    1  round   for  1  hour  consume 
300  bbls.  in  30  X  3  X  5  days. 
1  cannon  firing    1  round    for  1  hour  consumes 

300  bbls.  in  44  X  30  X  3  X  5  days. 
1  cannon  firing    1  round    for  1  hour  consumes 

300  •' 


teachers'  edition.  239 


66  cannon  firing    1  round    for  1  hour   consume 

300  X  66  ^ 

66  cannon  firing  40  rounds  for  1  hour   consume 

40  X  300  X  QQ       ^ 
66  cannon  firing  40  rounds  for  5  hours  consume 
Ibbhin    44X30X3X5 

5  X  40  X  300  X  66      ^ 
66  cannon  firing  40  rounds  for  5  hours  consume 

400  bbls.  in  400x44x30x3x_r>  ^       ^  ^  ^         ^^^ 
5  X  40  X  300  X  66         ^  ^ 

65.  How  many  times  will  a  wheel  2f  feet  in  circumference  turn 
round  in  travelling  over  12f  yards  ? 

The  wheel  revolves  once  in  going  2f-  feet. 

.-.  it  will  turn  around  as  many  times  in  going  12f  yds.  =  38f  ft. 
as  38f  ^  2f-  =  15.  Ans. 

66.  How  much  ground  will  be  travelled  over  by  a  wheel  If  yards 
in  circumference,  when  it  has  made  4|-  turns  ? 

If  the  wheel  turns  once  in  going  If  yds.,  in  making  4|  turns  it 
will  go  4^  times  If  yds.  =  6f  yds.  Ans. 

67.  Find  the  circumference  of  a  wheel  which  makes  9  turns  in 
travelling  over  7^  yards. 

If  it  makes  9  turns  in  going  7^  yds.,  it  will  go  7-^  yds.  -r-  9  =  f  yd. 

=  2f  ft.  in  making  one  turn. 
.•.  the  circumference  of  the  wheel  is  2|  feet.  Ans. 

68.  If  the  circumference  of  a  wheel  be  -^^^  of  1  yd.  li  ft.,  how  many 
times  will  it  turn  in  travelling  3f  miles  ? 

1  yd.  li  ft.  =  1-i  yds.  =  If  yds.       3f  mi.  =  3f  X  1760  yds. 

o 

If  the  wheel  makes  1  turn  in  going  -V_  of  If  yds.,  it  will  make  as 

many  turns  in  going  3f  X  1760  yds.  as  ^l^  ™^  =  UdQj\.  Ans. 


240  ARITHMETIC. 


69.  If  the  wheel  of  a  locomotive  be  3|  times  5.52  feet  in  circum- 
ference, how  many  times  does  it  turn  in  a  minute,  when  the  locomo- 
tive is  running  at  the  rate  of  13.34  miles  an  hour? 

5.52  =  5^1,  and  13.34  =  13^^. 

If  it  is  going  at  the  rate  of  13^^  miles  per  hour,  it  is  going  at  the 

rate  of  ^^  miles  per  minute,  or  at  the  rate  of  ^^H  X  ^^^^  feet 
60  ^  60 

per  minute. 
If  it  turns  once  in  going  3|  x  5^f  feet,  it  will  turn  as  many  times 

.       13Ux5280.    .       13Ux5280     /oi  .,  ki<.\     c-2 
in  going  —^^ feet  as  —^^ (H  X  5^1)  =  6/f . 

Ans. 


70.  A  can  run  /-  of  a  mile  in  |  of  a  minute,  B  can  run  /^  of  a 
mile  in  |  of  a  minute,  and  C  -^^  of  a  mile  in  f  of  a  minute.  Which 
is  the  fastest  runner?  and  if  he  can  run  a  certain  distance  in  3  min. 
10  sec,  how  much  longer  will  each  of  the  others  take  to  run  the 
same  distance  ? 

If  A  can  run  /^  mi.  in  f  min.,  to  run  a  mile  it  will  take  him  | 

min.  -^  "5^  =  8^  min. 
If  B  can  run  -^^  mi.  in  f  min.,  to  run  a  mile  it  will  take  him  | 

min.  ■^■i'z=Vj  inin- 
If  C  can  run  -^  mi.  in  f  min.,  to  run  a  mile  it  will  take  him  f 

min.  -4-  ^g  min.  =  6|^  min. 
.-.  C  is  the  fastest  runner. 
3  min.  10  sec.  =  3^  min. 
If  C  can  run  a  certain  distance  in  3^  min.,  and  a  mile  in  6}^ 

min.,  the  distance  is  that  part  of  a  mile  which  3^  is  of  6{^  =  ^. 
If  A  can  run  a  mile  in  8J  min.,  he  can  run  -j^  mi.  in  ^  of  8^ 

min.  ==  3^^  min.  =  3  min.  51  sec. 
But  C  can  run  it  in  3  min  10  sec.  .*.  it  takes  A  41  sec.  longer  than  C. 
If  B  can  run  a  mile  in  7^^  min.,  he  can  run  ^  mi.  in  /y  of  7-j^ 

min.  =  3i§|  =  3  min.  29^15  sec. 
But  C  can  run  it  in  3  min.  10  sec.    .*.  it  takes  B  19j^  sec.  longer 

than  C. 


TEACHERS     EDITION. 


241 


Find  the  amount  of  the  following  bills  : 

71. 

Mr.  Richard  Rowe,  Boston,  Nov.  23,  1880. 

To  John  Doe,  Dr. 


To  125  lbs.  sugar 

@  10  cts. 

"        1  hag  coffee,  115  lbs. 

@  32  cts. 

"      25  gals,  molasses 

@  62  cts. 

"        8  lbs.  Japan  tea 

@  92  cts. 

"      28  lbs.  crackers 

@,     8  cts. 

2  bbls.  flour 

@  $7.50 

$12 

36 

15 

7 

2 

15 


50 
80 
50 
36 
24 
00 


40 


Received  Payment, 


72. 


John  Doe. 


Mr.  James  Hardy,  Boston,  Feb.  29,  1888. 

To  a  H.  Mills,  Dr. 


To 


275  bbls.  flour      @  $6.75 

324  bbls.  flour      @  $6.25 

300  bu.  potatoes  @  48  cts 

1578  lbs.  butter      @  32  cts 

2000  bbls.  apples   @  $1.25 

1    car-load    oats,    20,000    lbs.,    625    bu. 

@  42  cts 

1  car-load  corn,  28,575  lbs.,  510.27  bu. 
(a),  55  cts 


$1856 

2025 

144 

504 

2500 

262 

280 


25 

00 
00 
96 
00 

50 

65 


Received  Payment, 
73. 


$7573    36 
a  IT.  Mills. 


James  Harlow, 


Boston,  Jan.  1,  1888. 
To  John  Dike,  Dr. 


To  12  bales  Texas  cotton,  5760  lbs. 

@ 

91  cts. .     . 

$532 

80 

"      7  bales  upland  cotton,  3514  lbs. 

@ 

10|  cts. .     . 

360 

19 

"      3  bales  low  middling,  1476  lbs. 

@ 

9f  cts. .     . 

143 

91 

"      8  bales  good  ordinary,  9220  lbs. 

@ 

9     cts..     . 

793 

80 

Received  Payment, 


$1830    70 
John  Pike. 


242 


ARITHMETIC. 


Exercise  LIX. 


1.  What  length  of  board  15 
in.  wide  will  contain  11  sq.  ft. 
36  sq.  in.  ? 

11  sq.  ft.  36  sq.  in.  =  llj  sq.  ft. 
15  in.  =  IJ  ft. 
llj  -^  li  =  9  ft.  Ans. 

2.  What  length  of  road  44  ft. 
wide  will  contain  an  acre  ? 

1  A.  =  160  sq.  rds. 
44  ft.  =  2|  rds. 
160  -  2i  =  60  rds.  Ans. 


3.  Find  the  area  of  a  rectan- 
gular field  13.12  chains  long, 
10.35  chains  broad. 

13.12  ch. 
X  10.35  ch. 

10)135.792  sq.  ch. 

13  A.  5.792  sq.  ch.  Ans. 

4.  A  path  216  ft.  long  meas- 
ured 72  sq.  yds.   Find  its  breadth . 

72  sq.  yds.  =  648  sq.  ft. 
648  +  216  =  3  ft.  Ans. 

5.  A  rectangular  field  of  21.66 
acres  is  250.8  yds.  broad.  Find 
its  length. 

1  A.  =  4840  sq.  yds. 

lM2^2L66.418yds.^ns. 
250.8  ^ 


6.  What  is  the  area  of  a  table 
if  length  and  breadth  be  4  ft.  3^ 
in.  and  2  ft.  9|  in.,  respectively  ? 

4  ft.  3f  in.  =  ^  ft. 
2  ft.  9f  in.  =  2f  ft. 

4f  X  2f  =  12  sq.  ft.  Ans. 

7.  From  each  corner  of  a 
square,  the  side  of  which  is  2  ft. 
5  in.,  a  square  measuring  5  in. 
on  a  side  is  cut  out.  Find  the 
area  of  the  remainder  of  the  figure. 

2  ft.  5  in.  =  2^5^  ft. 
2Ax2/j  =  51Hsq.ft 

=  5  sq.  ft.  121  sq.  in. 
5  X  5  =  25  sq.  in. 
25  sq.  in.  X  4  =  100  sq.  in. 
5  sq.  ft.  121  sq.  in.  =  100  sq.  in. 
=  5  sq.  ft.  21  sq.  in.  Ajis. 

8.  The  length  and  breadth  of 
a  map  are  4 J  ft.  and  3^  ft.,  re- 
spectively. If  the  map  represent 
77,760  sq.  mi.  of  country,  how 
many  miles  are  there  to  a  square 
inch? 

4^X3^  =  15  sq.ft. 

=  2160  sq.  in. 
77,760  +  2160  =  36  sq.  mi.  Ans. 

9.  In  rolling  a  grass  plot  24 
yds.  long  and  containing  400  sq. 
yds.,  how  many  times  must  a 
roller  3  ft.  4  in.  wide  be  drawn 


TEACHERS     EDITION. 


243 


over  it  lengthwise   so    that   the 
whole  may  be  rolled? 
400  -  24  =  16f  yds. 


10.    How  many  sods,  each  2 


ft.  3|-  in.  long  and  8|-  in.  broad, 
would  be  required  to  turf  an  acre 
of  ground  ? 

2  ft.  3|in.  =  27iin. 

1  A.  =  6,272,640  sq.  in. 
6272640 


27i  X  8^ 


=  27,648.  Ans. 


11.   Find  the  area  of  a  picture-frame  2^  in.  broad  and  having  an 
outside  measurement  of  4  ft.  6|  in.  in  length  and  2  ft.  8  in.  in  width. 


to, 


CC| 

^■1 


21  in.  X  2  =  4i  in. 
4  ft.  6|  in.  -  41  in.  =  4  ft.  2  in. 
4  ft.  2  in.  +  2  ft.  8  in.  =  6  ft.  10  in 
6  ft.  10  in.  X  2  =  13  ft.  8  in.  =  1 3f  ft. 
2iin.  =  fVft. 

P        lb       lb 

^T%  ^q-  ft-  =  -  ^c[.  ft.  81  sq.  in.  Ans 


4  ft.  2  in. 


12.  Find  the  expense  of  glazing  four  windows,  each  containing 
12  panes,  the  panes  being  each  a  foot  long  and  10  in.  wide,  and  the 
price  of  the  glass  38  cents  per  square  foot. 


10  in.  =  f  ft 
2      n 


1  X  f  =  I-  sq.  ft. 
XiiX-  =  10  sq.  ft.  10  sq.  ft.  X  4  =  40  sq.  ft. 

40x10.38  =  $15.20.  Ans. 


13.  A  garden  76  yds.  long  and  56  yds.  broad,  enclosed  by  a  wall, 
has  a  border  4  ft.  wide  within  the  wall,  and  within  this  a  ])ath  5  ft. 
wide,  the  middle  being  grass.  Find  the  areas  of  the  border,  path, 
and  grass,  respectively. 

4  ft.  =  li  yds.     2  X  1^  yds.  =  2f  yds. 
76  yds.  -  2f  yds.  =  73i  yds. 
731  yds.  +  56  yds.  =  1291  yds. 

1291  yds.  X  2  =  258 1  yds.  =  perimeter. 
258§  X  li  =  H-  X  i  =  ^^^  =  344|  sq.  yds.  (1)  Ans. 


244 


ARITHMETIC. 


■7^  y^s-  56  yds.  -  2f  yds.  =  53^  yds. 

731  yds.  -  2  X  If  =  70  yds. 
70  yds.  +  53|  yds.  =  123^  yds. 
123^  yds.  X  2  =  246f  yds. 
246fxlf  =  ^Xi 
=  ^^ 

=  411^Bq.  yds. 
(2)  Ans. 

5  ft.  +  4  ft.  =  9  ft.  =  3  yds. 
76  yds.  -  (2  X  3  yds.)  =  70  yds. 
56  yds.  -  (2  X  3  yds.)  =  50  yds. 

70  X  50  =  3500  sq.  yds.  (3)  Am. 


il 

Ur            73|- 

g 

en 

*           70              1 

1 

i 

i 

1 

14.  Find  the  area  of  a  circle 
which  has  a  radius  of  3  ft. 

3  X  3  -  9  sq.  ft. 
3.1416 

X  9  sq.  ft. 

28.27^^  sq.  ft.  Ans. 

15.  What  is  the  area  of  a  cir- 
cular field  with  a  radius  of  400 
yards  ? 

400  X  400  =  160,000  sq.  yds. 
3.1416 
X  160000  sq.  yds. 

502,656  sq.  yds.  Ans. 

16.  The  radius  of  the  rotunda 
of  the  Pantheon  at  Rome  is  71  ft. 
6  in.     Find  the  area  of  the  floor. 

71  ft.  6  in.  -  7H  ft. 
71ix71i  =  5112^  sq.ft. 
3.1416 
X5112^  sq.  ft. 

16060.64;IP  sq.  ft.  Ans. 


17.  The  diameter  of  a  cistern 
is  13  ft.  What  is  the  area  of  the 
bottom  ? 

13  ft.  -^-  2  =  Gl-  ft.  Radius. 
6^X6^  =  42^  sq.  ft. 
3.1416 
X42^  sq.  ft. 

i32.'/32^  sq.  ft.  Ans. 

18.  The  two  dials  of  the  clock 
of  St.  Paul's,  London,  are  each 
18}  feet  in  diameter.  What  is 
the  area  of  each  in  square  feet  ? 

18}  ft.  +  2  =  9^j  ft. 
9i^X  9,^  =  82^^  sq.ft. 
3.1416 

_J<82^^  sq.  ft. 

258.52jr^  sq.  ft.  Ans. 

19.  How  many  square  inches 
on  the  surface  of  a  ball  3  in.  in 
diameter  ? 

3  X  3  =  9  sq.  in. 
3.1416 

X  9  sq.  in. 

28.27)1)1  sq.  in.  Ana. 


TEACHERS     EDITION. 


245 


20.  How  many  square  inches 
of  surface  in  a  spherical  black- 
board 12  in.  in  diameter? 

12  X  12  =  144  sq.  in. 
3.1416 
X  144  sq.  in. 

452.39^^  sq.  in. 

21.  What  is  the  interior  sur- 
face of  a  hemispherical  vase  20 
in.  in  diameter? 

20  X  20  =  400  sq.  in. 
400  sq.  in.  -J-  2  -  200  sq.  in. 
3.1416 

X  200  sq.  in. 

628.32  sq.  in. 
=  4  sq.  ft.  52.32  sq.  in. 

22.  How  many  yards  of  car- 
peting f  of  a  yard  wide  will  be 
required  for  a  floor  26  ft.  long, 
15|  ft.  wide,  if  the  strips  run 
lengthwise?  How  many  if  the 
strips  run  across  the  room? 
How  much  will  be  turned  under 
in  each  case  ? 

15f  ft.  =  51  yds. 
5J  -5-  f  =  7  strips. 
26  ft.  =  8|  yds. 
7  X  8f  yds.  =  60f  yds.  Ans. 
None  to  turn  under. 
8f-v-f=llf=  12  strips. 
12  X  5|  yds.  =  63  yds.  Ans. 

4     3     1 

^  X  -  =  -  yd.  to  turn  under, 
^      4      o 


23.  How  many  yards  |  of  a 
yard  wide  will  be  required  for 
a  room  8|-  yds.  long  and  17  ft. 
wide,  if  the  strips  run  length- 
wise, and  there  is  a  waste  of  j^^- 
of  a  yard  in  each  strip,  in  match- 
ing patterns  ? 

17  ft.  =  5f  yds. 
5f  -  I  =  6^f  =  7  strips. 

7  X  81  yds.  -  59i  yds. 
59^-  yds.  +  j\  yds. 

=  59{^jds.  Ans. 

24.  How  many  square  yards 
of  oil-cloth  will  be  required  for 
a  hall  floor  6^  yds.  long  and 
10  ft.  wide  ? 

10  ft.  =  31  yds. 
5|  X  3 1  =  17|sq.yds.  Ans. 

25.  What  will  be  the  cost  of 
carpet  |  of  a  yard  wide  for  a 
room  28 1  ft.  by  18|  ft.,  if  the 
strips  run  lengthwise,  and  the 
cost  per  yard  is  92  cents  ? 

18f  ft.  =  6|  yds. 
6^  -^1=5  strips. 
28|~  ft.  =  9^  yds. 
91  X  5  =  471  yds. 
471  X  $0.92  =  $43.70.  ^^s. 

26.  Find  the  cost  of  carpet 
30  in.  wide,  at  $1.25  per  yd.  for 
a  room  18  ft.  by  14  ft.,  if  the 
strips  run  lengthwise ;  if  the 
strips  run  across  the  room. 


246 


ARITHMETIC. 


30  in.  =  §  yd. 
18  ft.  =  6  yds. 
14  ft.  =  4f  yds. 
4|  -5-  f  =  5f  =  6  strips. 
6x6  yds.  =  36  yds. 
36x$U  =  $45.  Ans. 
6  -  f  =  7^  =  8  strips. 
8x4f  x$li  =  $46| 
=$46.67.  Ans. 

27.  Find  the  cost  of  carpeting 
27  inches  wide,  at  $1.12^  per 
yard,  for  a  room  29  ft.  9  in.  by 
23  ft.  6  in.,  if  the  strips  run 
across  the  room. 

27  in.  =  f  yd. 
29  ft.  9  in.  =  29 J  ft.  =  9}^  yds. 
23  ft.  6  in.  =  23^  ft.  =  7f  yds. 
9H-^f  =  13f. 
14  X  7f  =  109|  yds. 
=  14  strips. 
109fx$  1.125  =  $123.38.  Ans. 

28.  Find  the  cost  of  carpeting 
f  of  a  yard  wide,  at  $2.75  per 
yard,  for  a  room  34  ft.  8  in.  by 
13  ft.  3  in.,  if  the  strips  run 
lengthwise,  and  if  there  be  a 
waste  of  ^  of  a  yard  on  each 
strip  in  matching  the  pattern. 

34  ft.  8in.  =  34Jft.  =  ll|yds. 
13  ft.  3in.  =  13jft.  =  4-i5jyd8. 
4fk  -!- 1  =  5J  =  6  strips. 
6  X  n  fj  =  69}  yds. 
6  X  i  yd.  =  1^  yds.  waste. 
69i  +  H=.70tyd8. 
70JX  $2.75  =  $194.79.  Ans. 

29.  Which  way  must  the 
strips  of  carpet  J  of  a  yard  wide 


run  in  order  to  carpet  most 
economically  a  room  20  ft.  6  in. 
long  and  19  ft.  6  in.  wide,  if 
there  be  no  waste  for  matching 
the  pattern  ? 

20  ft.  6  in.  =  20^  ft.  =  6f  yds. 
19  ft.  6  in.  =  19^  ft.  =  6^  yds. 

6|  -  f  =  8f  =  9  strips. 

9  X  6f  =  61^  yds.  lengthwise. 

6f  -=-  f  =  9^  =  10  strips. 
10  X  6|  =  65  yds.  across. 
.•.  the  strips  must  run  lengthwise. 

30.  Find  the  number  of  yards 
of  plastering  in  the  walls  of  a 
room  2 If  ft.  long,  16^  ft.  wide, 
and  11  ft.  high,  if  12  sq.  yds.  be 
allowed  for  doors,  windows,  and 
base-boards. 

21|  +  16^  =  38^ft. 
2  X  38^  ft.  ==  76^  ft. 
76^X11  =  841^  sq.ft. 
=  93^  sq.  yds. 
93^  sq.  yds.  —  12  sq.  yds. 

=  81^  sq.  yds.  Ans. 

31.  How  many  square  yards 
of  plastering  in  the  walls  and 
ceiling  of  a  room  30  ft.  8  in.  long, 
26  ft.  5  in.  wide,  10  ft.  6  in.  high, 
if  24  sq.  yds.  be  allowed  for  doors, 
windows,  and  base-boards  ? 

30  ft.  8  in.  =  30f  ft. 
26  ft.  5  in.  =  261&J  ft. 
30J  -f-  26t\  =  57tV  ft. 
2x57T»j  =  114ift. 
114J  xlOi  =  133^5  sq.  yds. 
30Jx2635j  =  905VBq.yd8. 
133,V  +  90^  =  223j^. 
223^-24=199^ sq. yds.  Ans. 


TEACHERS     EDITION. 


247 


32.  What  will  be  the  cost  of 
plastering  the  walls  and  ceiling 
of  a  room  27  ft.  4  in.  long,  20  ft. 
wide,  and  12  ft.  6  in.  high,  at  27 
cents  per  square  yard,  if  20  sq. 
yds.  be  deducted  for  doors,  win- 
dows, and  base-board  ? 

27  ft.  4  in.  =  27i  ft.  =  9^  yds. 

20  ft.  =  6f  yds. 
12  ft.  6  in.  =  12i  ft.  =  4i  yds. 
9i  +  6f  =  15|yd8. 
2xl5|  =  31|yds. 
31f  X4i  =  131if  sq.yds. 
9-1  X  6f  =  60|^  sq.  yds. 
131|f  X  60f  ^  =  192f  sq.  yds. 
192f-20=172|8q.  yds. 
172f  x|0.27  =  |46.50.  Ans. 

33.  Find  the  cost  of  whitening 
the  ceiling  and  walls  of  a  room 
14  ft.  4  in.  wide,  15  ft.  6  in.  long, 
10  ft.  6  in.  high,  at  5  cents  per 
square  yard,  allowing  9  sq.  yds. 
for  doors  and  windows. 


ft.  4  in.  =  14-1-  ft. 

=  4|  yds. 

ft.  6  in.  =  15i  ft. 

=  5iyds. 

ft.  6  in.  =  10|  ft. 

=  H  yds. 

4|  +  5i  =  9Hyd8. 

2x9i|=19|yds. 

19|x3i  =  69iisq.yds. 

4|  X  51  =  24||  sq.  yds. 

69ii  +  24f|  =  94^Vsq.yds. 

942V  -  9  =  85387  sq.  yds. 

85^^  X  ? 0.05  =  $  4.26.    Ans. 

34.  Find  the  cost  of  plastering 
a  room  21  ft.  long,  15  ft.  wide, 
12  ft.  high,  at  40  cents  per  square 
yard,  allowing  for  a  door  7  ft. 
high,  3  I  ft.  wide;  3  windows, 
each  5  ft.  high,  3  ft.  wide ;  and 
a  dado  2  ft.  9  in.  high  around  the 
room. 

21  -f  15  =  36  ft. 
2  X  36  ft.  =  72  ft. 

72x12  =  1179  sq.ft. 
3  X  7  =  21  sq.  ft.  door. 
5  X  3  X  3  =  45  sq.  ft.  windows. 

2|  X  72  =  198  sq.  ft.  dado. 

21  -f  45  -f  198  =  264  sq.  ft. 

1179  -  264  =  915  sq.  ft. 
915  sq.  ft.  =  101f  sq.  yds. 
lOlfx  $0.40  =  140.67.  Ans. 


35.  Find  the  cost  of  papering  a  room  20  ft.  6  in.  long,  17  ft.  4  in. 
wide,  9  ft.  high,  with  paper  18  in.  wide,  8  yards  in  a  roll,  at  75  cents 
a  roll ;  allowing  for  2  doors,  each  7  ft.  high,  3  ft.  wide,  and  for  3 
windows,  each  5  ft.  6  in.  high  and  3  ft.  3  in.  wide. 

20  ft.  6  in.  =  20i  ft.  =  6f  yds. 
17  ft.  4  in.  =  17i  ft.  =  51  yds. 
9  ft.  =  3  yds. 
^  +  5l  =  l2\ljds.     2xl2ii  =  25f. 


2  x  7  X  3  =  42  sq.  ft.  =  4f  sq.  yds.  doors. 


248  ARITHMETIC. 

3  X  5|  X  3|  =  53|  sq.  ft.  =  5f  f  sq.  yds.  windows. 
4f +  5||=10f  sq.  yds. 
75f  -  lOf  =  65^5  sq.  yds. 

18  in.  =  ^  yd.     8x^  =  4  8q.  yds. 
65^5  ^  4  =  16f|  =  17  rolls. 
17  x!p0.75  =  $  12.75.  Ans. 

36.  Find  the  cost  of  papering  a  room  32  ft.  long,  22  ft.  wide,  13  ft. 
high,  with  paper  18  in.  wide,  8  yards  in  a  roll,  at  $1.25  a  roll,  if 
50  sq.  yds.  be  allowed  for  doors,  windows,  and  base-board. 

32  +  22  =  54  ft.    2  X  54  =  108  ft. 
108  X  13  =  1404  sq.  ft.  =  156  sq.  yds. 
156  -  50  =  106  sq.  yds. 

18  in.  =  ^  yd.     8x^  =  4  sq.  yds. 
106  -^  4  =  26^  =  27  rolls. 
27  X  $1.25  =  $33.75.  Ans. 

37.  Find  the  cost  of  papering  a  room  26  ft.  long,  21  ft.  wide,  12 
ft.  high,  with  paper  20  in.  wide,  8  yards  in  a  roll,  at  $1.50  a  roll, 
and  a  border  at  25  cents  per  running  foot;  allowing  for  a  fire-place 
6  ft.  3  in.  by  4  ft.,  a  door  7  ft.  by  4^  ft.,  and  3  windows,  each  6  ft. 
by  3i  ft. 

26  ft.  =  8f  yds.     21  ft.  =  7  yds.     12  ft.  =  4  yds. 
8f  +  7  =  15|yd8.     2xl5f  =  3Hyds. 
3H  X  4  =  125}  sq.  yds. 
5  ft.  3  in.  =  5^  ft. 

5^  X  4  =  21  sq.  ft.    =   2J  sq.  yds.  fire-place. 
7  X  4i  =  31}  sq.  ft.  =    3}  sq.  yds.  door. 
3  X  6  X  3}  =  63  sq.  ft.    =    7    sq.  yds.  windows. 

12|  sq.  yds.  to  be  deducted. 

125}-12J  =  112}8q.  yds. 

20  in.  =.  f  yd.     8  x  f  =  4f  sq.  yds. 
112i-4-4J  =  25T!V  =  26roll8. 
26  x$  1.50  =  $39.00. 
Perimeter  =  31}  yds.  =  94  ft. 
94  X  $0.25  =  $23.50. 
$39.00  +  $23.50  =  $62.50.  Am. 


teachers'  edition.  249 

How  many  feet  board  measure  in  : 

38.  A  board  18  ft.  long,  9  in.  wide,  |  in.  thick? 
A  board  16  ft.  long,  11  in.  wide,  1  in.  thick  ? 

9  in.  -  f  ft. 
18xf  =13ift.  Ans. 
11  in.  =  |i  ft. 
16  X  H  -  14f  ft.  Ans. 

39.  Twenty  boards  averaging  14  ft.  long,  10  in.  wide,  ^  in.  thick? 

10  in.  =  I  ft. 

Mx^x22  =  ^  =  233ift.  ^ns. 
1^13  ^ 

3 

40.  Three  joists  13  ft.  long,  8  in.  wide,  3  in.  thick? 

8  in. -f  ft. 
Hence  1  joist  =  3  boards  13  by  f. 

^x|xfxf=78ft.  Ans. 
1      p      1      1 

41.  A  stick  of  timber  8  in.  by  9  in.  and  27  ft.  long? 

8  in.  ==  I  ft. 

Hence  1  stick  =  9  boards  27  ft.  by  f  ft. 

9 

^x|x?  =  162ft.  Ans. 
1       P      1 

42.  Two  beams,  each  6  in.  by  9  in.  and  23  ft.  long  ? 

6  in.  -  i  ft. 
Hence  1  beam  =  9  boards  23  ft.  by  |  ft. 

^xix^X^-207ft.  Ans. 
1/11 


250  ARITHMETIC. 


43.   Three  joists,  each  3  in.  by  4  in.  and  11  ft.  long' 


Hence  1  joist  =  4  boards  11  ft.  by  \  ft. 


i^xix^X?  =  33ft.  Ans. 
1^11 


44.  Five  joists,  each  6  in.  by  4  in.  and  14  ft.  long? 

6  in.  =  I  ft. 

Hence  1  joist  =  4  boards  14  ft.  by  ^  ft. 

2 

i^  X  J  X  f  X  ^  =  140  ft.  Ans. 
1       /S      1      1 

45.  A  stick  of  timber  10  in.  square  and  36  ft.  long  ? 

10  in.  =  f  ft. 
Hence  1  stick  =  10  boards  36  ft.  by  f  ft. 


^x|x^  =  300ft.  Ans. 
1      p       1 


46.   Ten  planks,  each  13  ft.  long,  15  in.  wide,  2  in.  thick? 

15  in.  ==  f  ft. 
Hence  1  plank  =  2  boards  13  ft.  by  f  ft 

^X^x|x^  =  325ft.  Ans. 
14     11 


Find  the  cost  of: 

47.  Nine  joists,  each  15  ft.  long,  3 J  in.  by  5  in.,  at  fil2  per  M. 

5  in.  -  -jf^  ft.    1 12  per  M.  =  -^  per  sq.  ft. 

Hence  1  joist  =■  3^  boards  15  ft.  by  -j^  ft. 


teachers'  edition.  251 

48.   Thirty  planks,  each  12  ft.  long,  11  in.  wide,  3  in.  thick,  at 
$  15  per  M. 

llin.  =  ilft.     |15perM.--^~  per  sq.ft. 

Hence  1  plank  =  3  boards  12  ft.  by  |^  ft. 

12xlix3x30x^  =  l|l  =  fU.85.^«. 


49.   Four  sticks  of  timber,  each  8  in.  by  9  in.  and  23  ft.  long,  at 
|18per  M. 

9in.  =  fft.    $  18  per  M.=  11^  per  sq.ft. 

Hence  1  stick  =  8  boards  23  ft.  by  |  ft. 

T>^4'^l>'l''l000-"T25~-^^-^^-  ^^'- 


50.   A  board  24  ft.  long,  23  in.  wide  at  one  end  and  17  in.  at  the 
other,  and  1^  in.  thick,  at  $30  per  M. 

^^"^•^'^  =  20  in.  average  width. 

20  in.  =  If  ft 

$30  per  M.  =  ^  per  sq.ft. 

Hence  the  board  =  1^  boards  24  ft.  by  If  ft, 

24x-X-X-^  =  ^  =  $1.80.  Ans. 
3     2      1000       5       * 


51.  A  stick  of  timber  29  ft.  long,  10  in.  by  12  in.  at  $  13.50  per  M. 

12  in.  =  1  ft.  $  13.50  per  M.  =  tiM  =  31L  per  ft. 
^  1000      2000^ 

Hence  1  stick  =  10  boards  29  ft.  by  1  ft. 

29  X  ;^  X  1  X  ^^  =  ^^  =  $3.92.  Ans. 
200P      200       * 

52.  The  flooring  for  two  floors,  each  23  ft.  by  17  ft.,  each  floor 
double,  and  of  boards  ^  in.  thick ;  the  lower  floor  at  $  18,  and  the 
upper  at  $  24,  per  M. 


252 


ARITHMETIC. 


782  sq.  ft. 


Boards  |  in.  thick  are  reckoned  as  1  in. 
Hence  each  floor,  being  double,  will  require  2  X  23  X  17 
Average  cost  per  floor  =  $  21  per  M. 
.'.  Average  cost  both  floors  =  f  42  per  M. 
$42 


Whole  cost  =  782  x 


1000 


$32.84.  ^718. 


17  ft. 


53.  The  flooring  timbers  for  a  room  23  ft.  by  17  ft.  at  $18  per  M, 
if  they  are  2  in.  by  10  in.,  17  ft.  long,  and  are  placed  on  edge,  two 
close  to  the  walls,  and  the  others  with  spaces  of  f^  of  a  foot  between 
them. 

The  room  being  17  ft.  wide  and  the  timbers 
17  ft.  long,  the  timbers  must  run  across  the  room. 
After  a  timber  is  placed  against  the  wall  at  one 
end,  the  remaining  distance  to  be  occupied  with 
timbers  and  spaces  =  23  ft.  -  2  in.  =  22f  ft.  The 
distance    occupied    by   a    timber    and    a  space 

.-.  22f 
remaining  space. 

20 
137 


^f  ^  =  number  of  timbers  required  for 


22-1-^ 


120       ^ 


20. 


20  +  1 


1  timber  is  supposed  to  have  been  placed. 
21,  the  whole  number  of  timbers  required. 
3 


XPxixl7x 


$;^_$5i 


1000     100 
21  X  $0.51  =  $  10.71,  whole  cost. 


=  $0.51,  cost  of  one  timber. 


54.    A  log  14  ft.  long,  17  in.  in  diameter. 

17^-2x17  =  289-34  =  255. 
H  of  i^  of  255  =  187  ft.  Am. 


'5.   A  log  11  ft.  long,  13  in.  in  diameter. 

13'  -  2  X  13  =  169  -  26  =  143. 
fi  of  H  of  143  =  83  ft.  Aii%. 


teachers'  edition.  253 

56.  A  log  16  ft.  long,  20  in.  in  diameter. 

202  -  2  X  20  ==  400  _  40  =  360. 

21  of  ^  of  360  =  302  ft.  Ans. 

57.  A  log  12  ft.  long,  15  in.  in  diameter. 

152  -  2  X  15  =  225  -  30  =  195. 

2^  of  if  of  195  =  123  ft,  Ans. 

Find  the  value,  at  $  9  per  M.  of : 

58.  A  log  17  ft.  long,  averaging  11  in.  in  diameter. 

112-2x11  =  121-22  =  99. 

f ^  of  \i  of  99  =  88.3575  ft. 
88.3575  X  10.009  =  $0.80.  Ans. 

59.  A  log  18  ft.  long,  averaging  13  in.  in  diameter. 

132-2x13  =  169-26  =  143. 

I^ofif  of  143  =  135.135  ft. 
135.135x10.009  =  $1.22.  Ans. 

60.  A  log  13  ft.  long,  16  in.  in  diameter. 

162  -  2  X  16  =  256  -  32  =  224. 
1^  of  If  of  224  =  153  ft. 

153  X  $0,009  =  $1.38.  Ans.  ., 

61.  A  log  14  ft.  long,  12  in.  in  diameter. 

122  -  2  X  12  =  144  -  24  =  120. 
f^ofif  of  120  =  88ft. 

88  X  $0,009  =  $0.79.  Ans. 

62.  How  many  clapboards  will  be  required  to  cover  the  front  of 
a  house  60  ft.  long  and  20  ft.  high,  if  they  are  laid  4  in.  to  the 
weather,  and  if  120  sq.  ft.  be  deducted  for  doors  and  windows  ? 

60  X  20  =  1200. 
1200  -  120'=  1080. 

4  X  i  =  li  sq.  ft. 
1080  H- 11  =  I  of  1080. 
=  810.  Ans. 


254 


ARITHMETIC. 


63.  If  one  thousand  shingles  cover  120  sq.  ft.  of  roof,  what  is  the 
average  width  of  a  shingle  ? 

T'(ftp7  =  /:?8q-ft.  =  17273sq.  in. 
I  of  16  in.  -  5]  in. 
173;^-5i  =  3/^in.  Am. 

64.  Allowing  one  thousand  shingles  for  120  sq.  ft.,  how  many- 
thousand  will  be  required  to  cover  the  pitched  roof  of  a  house  60  ft. 
long,  if  the  width  of  each  side  of  the  roof  be  24^  ft.  ? 

2^  X  60  =  1470  sq.  ft. 
1470  V  120  =  24i 

Ans. 


1.   Find    the    volume    of    a 

rectangular  solid  whose  length, 

breadth,  and  thickness  are  7  ft, 

2  ft.  6  in.,  and  11  in.  respectively. 

7  X  2i  X  ii  =  le^j  cu.  ft. 

=>  16  cu.  ft.  72  cu.  in.  Ans. 


Exercise  LX. 

5.  IIow  many  cubic  feet  of 
water  does  a  cistern  hold  whose 
length,  breadth,  and  height  are 
5  ft.  4  in.,  3  ft.  6  in.,  2  ft.  10  in., 
respectively  ? 

5|  X  3i  X  2f  =  52|  cu.  ft.  Ans. 


2.  How  many  cubic  feet  of 
air  in  a  hall  54  ft.  long,  33  ft. 
wide,  21  ft.  4  in.  high  ? 

54  X  33  X  21|  =  38,016  cu.  ft. 
Ans. 

3.  Find  the  volume  of  a  cube 
whose  edge  is  2]^  yds. 

2i  X  2J  X  2i  =  15f  cu.  yds. 
=  15  cu.  yds.  16  cu.  ft. 
1512  cu.  in.  Ans. 

4.  A  cellar  is  dug  21  ft.  long, 
17  ft.  3  in.  wide,  9  ft.  deep.  How 
many  cubic  yards  of  earth  are 
taken  out  ? 

21X17^X9 
27 


120|cu.yd8.  Ans. 


6.  If  the  dimensions  of  a  brick 
be  8  in.  by  3^  in.  by  2^  in.,  find 
its  volume. 

8  X  3i  X  2^  =  63  cu.  in.  Ans. 

7.  In  a  bar  of  iron  21  ft.  long 
3  in.  wide,  2  in.  thick,  how  many 
cubic  inches  are  there  ? 

21  ft.  =  252  in. 
252  X  3  X  2  =  1512  cu.  in.  Ans. 

8.  What  is  the  value  of  a  bar 
of  gold  8  in.  long  and  |  of  an 
inch  square,  at  |190  a  cubic 
inch? 

8xfXfX|190.  =  $855.  Ans. 


TEACHERS     EDITION. 


255 


9.  A  reservoir  whose  length 
and  breadth  are  15  yds.  and  12 
yds.,  respectively,  holds  330  cu. 
yds.  of  water.    What  is  its  depth  ? 

330 


15x12 


1|  yds.  Ans. 


10.  What  length  must  be  cut 
off  a  beam  9  in.  by  15  in.  to  con- 
tain 21  cu.  ft.  ? 


^ 


foff 


2fft. 
2  ft.  ) 


Ans. 


11.  How  high  should  a  room 
be  made,  if  its  length  be  31  ft. 
3  in.  and  breadth  24  ft.,  in  order 
that  it  may  contain  10,000  cu.  ft. 
of  air? 


10000 


=  13i  ft.  Ans. 


12.  A  piece  of  wood  5  ft.  long, 
1  ft.  broad,  and  9  in.  thick,  is  cut 
up  into  matches  2|  in.  long  and 
0. 1  of  an  inch  square.  How  many 
will  there  be  if  no  allowance  be 
made  for  waste  in  cutting? 

5  ft.  =  60  in. ;  1  ft.  =  12  in. 

60  X  12  X  9 


^  X  tV  X  tV 


=  259,200.  Ans. 


13.  How  long  a  wall  6  ft. 
high,  12J  in.  thick,  could  be  built 
with  the  bricks  forming  a  pile 
17  ft.  6  in.  long,  5  ft.  wide,  4  ft. 
3  in.  high  ? 


12|  in.  =  IxV  ft. 


17ix5x4 


f  =  581  ft.  Ans. 


14.  Find  the  surface  of  a  cube 
whose  edge  is  3  ft.  5|  in. 

5f  in.  =  i|  ft. 
3Hx3Hx6=72^^sq.ft. 
=  72  sq.  ft.  48f  sq.  in.  Ans. 

15.  Find  the  surface  of  a  rec- 
tangular block  of  stone  4  ft.  long, 
2|  ft.  broad,  1\  ft.  thick. 

4  X  2^  X  2  =  20. 
2|xlix2  =  6i. 
4  X  H  X  2  =  10. 
20  -f-  6^  +  10  =  36^  sq.  ft. 
=  36  8q.  ft.  36  sq.  in.  Ans. 

16.  A  lake  whose  area  is  45 
A.  is  covered  with  ice  3  in.  thick. 
Find  the  weight  of  the  ice  in 
tons,  if  a  cubic  foot  weigh  920  oz. 
avoirdupois. 

1  A.  =  43,560  sq.  ft. 
43560 
X45 
1960200  sq.  ft. 

3  in.  =  I:  ft. 
4)1960200 

490050  cu.  ft. 
X920 
450846000 

32,000  oz.  =  1  t. 

1408811 1.  Ans. 
32)450846 


256 


ARITHMETIC. 


17.  How  many  bricks  will  be 
required  to  build  a  wall  75  ft. 
long,  6  ft.  high,  and  16  in.  thick, 
each  brick  being  8  in.  long,  4  in. 
wide,  2J  in.  thick  ? 

75X6XH^14  4QQ^^ 

fxixA 


18.  Find  the  cost  of  making  a 
road  110  yds.  in  length  and  18 
ft.  wide,  the  soil  being  first  re- 
moved to  the  depth  of  1  ft.  at  a 
cost  of  25  cents  a  cubic  yard  ; 
rubble  being  then  laid  8  in.  deep, 
at  25  cents  a  cubic  yard,  and 
gravel  placed  on  top  9  in.  thick, 
at  62J  cents  a  cubic  yard. 


18  ft.  -  6  yds. 

110^6      1     $25^ 
1    ^1^3^  100~ 

8  in.  =  f  yd. 

110^6     2     $25_ 

1    ^1^9^100" 


$55. 


$36|. 


9  in. 


iyd. 


1  "^i^'i^'Tooo-^^^^*- 


Whole  cost  =  $  194.79. 
Ans. 

19.  A  room  whose  length  is 
27  ft.,  breadth  24  ft.,  height  10 
ft.,  is  to  have  its  ceiling  raised  so 
as  to  increase  the  space  by  84  cu, 
yds.  What  will  then  be  its 
height? 


27  ft.  =  9  yds. 
24  ft.  =  8  yds. 


9x8        '^ 
10  +  3|  =  13^  ft.  Ans. 


3nt. 


20.  A  block  of  wood  5  ft.  4.8 
in.  long,  1  ft.  9  in.  wide  and 
thick,  weighs  7.56  cwt.  Deter- 
mine the  weight,  in  pounds,  of  a 
cubic  foot, 

5  ft.  4.8  in.  =  5f  ft. 
7.56  cwt.  =  756  lbs. 


756 


5f  xlfxlf 


=  45^  lbs.  Ans. 


21.  How  many  cords  in  a  pile 
of  wood  40  ft.  long,  4  ft.  wide,  5 
ft.  4  in.  high  ? 

40X4X5^^3     ^^^^ 
8x4x4         * 


22.  A  pile  of  wood  containing 
67*  cords  is  270  ft.  long  and  4  ft. 
wide.     How  high  is  it? 


67^  X  128 
270x4 


8  ft.  Ans. 


23.  What  will  be  the  cost  of  a 
pile  of  wood  25  ft.  long,  4  ft. 
wide,  4  ft.  8  in.  high,  at  $3.75  a 
cord? 

25x4x4?x$3|^^^3^^^^ 
128 


TEACHERS     EDITION. 


257 


24.  What  must  be  the  length 
of  a  load  of  wood  3|  ft.  high  and 
5  ft.  wide,  to  contain  a  cord  ? 


128     ^128x2 
5x3i       5x7 


7Hft.  Ans. 


25.  How  high  must  manure 
be  in  a  cart  6  ft.  by  4  ft.,  in  order 
to  be  J  a  cord. 

k>il^  =  22  ft.  Ans. 
6x4         ^ 


26.  Find  the  number  of  bush- 
els in  a  bin  that  is  8  ft.  long,  4  ft. 
wide,  3  ft.  deep. 


X4x3 


96-4  of 


96  - 19^  -  76f  bu. 
Ans. 


27.  Find  the  number  of  bush- 
els in  a  bin  9  ft.  long,  6  ft.  6  in. 
wide,  3  ft.  4  in.  deep. 

9  X  6i  X  3i  =  195. 
195-^  of  195=195-39 

=  156  bu.  Ans. 


28.  Find  the  depth  of  a  bin 
to  hold  360  bu.,  if  its  length  be 
12  ft.  and  its  width  6  ft. 

360  -  i  of  360  =  450  cu.  ft. 
450 


12x6 


Q\it.  Ans. 


29.  Find  the  length  of  a  bm 
that  is  6  ft.  wide  and  5  ft.  deep, 
if  it  hold  400  bu. 

400  +  \oi  400  =  500  cu.  ft. . 

^^  =  ^-^  =  ^^  =  W^itAns. 
6x5      30       3  ^ 


30.  Find  the  number  of  bush- 
els that  will  fill  a  bin  8.5  ft.  long, 
4.5  ft.  wide,  3.5  ft.  deep. 

8.5  X  4.5  X  3.5  =  133.875  cu.  ft. 
133.875-^  of  133.875 

=  107.1  bu.  Ans. 


31.  A  bin  20  ft.  long,  12  ft 
wide,  and  6  ft.  deep,  is  full  of 
wheat.  What  is  its  value,  at 
11.25  a  bushel? 

20  X  12  X  6  =1440  cu.  ft. 
1440-1  of  1440  =  1152  bu. 

1152x|li=$1440.  Ans. 

32.  If  a  ton  of  coal  occupy  40 
cu.  ft.,  how  many  tons  will  a  bin 
hold  that  is  21  ft.  long,  10  ft. 
wide,  5  ft.  deep  ? 

21  X  10  X  5 


40 


26^  t.  Ans. 


33.  If  a  ton  of  Lehigh  coal 
occupy  35  cu.  ft.,  how  many  tons 
will  a  bin  hold  that  is  8  ft.  long, 
5  ft.  9  in.  wide,  4  ft.  6  in.  deep  ? 


sxjfxii 

35 


m  t.  Ans. 


258 


ARITHMETIC. 


34.  Find  the  number  of  gal- 
lons that  a  cistern  will  hold  that 
is  13  ft.  long,  6  ft.  wide,  7  ft.  4  in. 
deep. 

13  X  6  X  7J  =  572  cu.  ft. 

-  988,416  cu.  in. 

4278  f  gals.  Ans. 
231)988416 

35.  Find  the  number  of  gal- 
lons that  a  tank  will  hold  that  is 
4  ft.  long,  2  ft.  8  in.  wide,  1  ft.  8 
in.  deep. 

4x2|xl§  =  17^cu.  ft. 

1728 
XlT^ 
23l)30720(l32|f  gala.  Ans. 


36.   Find  the  number  of  gal- 
lons in  a  cubic  foot. 

7^}  gals.  Ans. 
23l)l728 


37.  Find  the  capacity  of  a 
cistern,  in  cubic  feet,  that  will 
hold  200  barrels  of  water. 


200  x  3H 
7i 


840  cu.  ft.  Am. 


38.  Find  the  number  of  gal- 
lons that  a  round  cistern  will 
hold  that  is  6  ft.  in  diameter  and 
7  ft.  deep. 


Z^  =  9. 
3.1416  X  9  X  7  X  7J 

=  1484.41  gals.  Ans. 

39.  Find  the  number  of  gal- 
lons that  a  vessel  will  hold  that 
is  12  in.  in  diameter  and  10  in. 


6in.  =  Ht.     (J)»  =  J. 
10  in.  =  6  ft. 
Jxfx3.1416x7J  =  4.91  gals. 
Ans. 


40.  How  many  quarts  will  a 
round  vessel  hold  5^  in.  in  diam- 
eter and  6  in.  deep  ? 

(5^)2  =  26M. 

0.7854 
X26ft 

20.9658 
X6 


126.7948  cu.  in. 

30  qts.  =  1  cu.  ft. 
••  1  qt.  =  jV  of  1728  cu.  in. 
=  57.6  cu.  in. 

2.18  qts.  Ans. 


576)1257.95 


41.  Find  the  number  of  cubic 
inches  in  a  sphere  11  in.  in  diam- 
eter. 

11  X  11  X  11  X  0.5236 

—  696.9  cu.  in.  Ans. 


TEACHERS     EDITION. 


259 


42.  How  many  quarts  will  a 
sphere  hold  that  is  12  in.  in  diam- 
eter ? 

13=1. 
0.5236  X  1  -  0.5236  cu.  ft. 

30  qts.  =  1  cu.  ft. 
0.5236  X  30  =  15.708  qts.  Ans. 

43.  What  part  of  a  bushel 
will  a  hemispherical  bowl  hold 
that  is  13  in.  in  diameter  ? 

13^  =  13x13x13  =  2197. 

0.5236 
X2197 


2)1150.3492  cu.  in. 
575.1746  cu.  in. 

1  bu.  =  2150.42  cu.  in. 

0.267.  Ans. 
215042)57517.460 

44.  If  a  cubical  box  2  ft.  on 
an  edge  contain  a  solid  sphere  2 
ft.  in  diameter,  how  many  gal- 
lons of  water  can  be  poured  into 
the  box  ? 

2x2x2  =  8. 
8x0.5236  =  4.1888. 

8.0000 

4.1888 


3.8112  cu.  ft. 
X1728 


6585.5536  cu.  in. 

28.51  gals.  Ans. 
231)6585.55 


45.  If  64  qts.  of  water  be 
poured  into  a  vessel  that  will 
hold  2  bu.  of  wheat,  what  part 
of  the  vessel  will  be  filled  ? 

2  bu.  =  64  dry  qts. 
64  dry  qts.  =  64  X  67^  cu.  in. 
64  liquid  qts.  =  64  X  57f  cu.  in. 

6i><i!7|  ^55  =  0.859.  ^m. 
64  X  67i      '^ 


46.   Find  the  number  of  cubic 
inches  in  1  oz.  (av.)  of  water, 

1000)1728.000 

1.728  cu.  in.  Ans. 


47.   Find  the  weight  in  ounces 
(av.)  of  1  cu.  in.  of  water. 

mt^Mfoz.  Ans. 


48.   Find  the  weight  in  ounces 
(av.)  of  1  pt.  of  water. 


1000 

7ix8 


16f  oz.  Ans. 


49.   Find  the  number  of  pints 
in  1  lb.  of  water. 

1  pt.  of  water  weighs  16|  oz. 
1  lb.  =  16  oz. 
16 


16^ 


=  11  pts.  Ans. 


.260  ARITHMETIC. 


50.  Find  the  weight,  in  grains,  of  1  cu.  in.  of  water. 

1         ■       c      ,  ■  I.    1000  1000      1, 

1  cu.  in.  of  water  weighs oz.  or lbs. 

^      1728  1728  X  16 

51.  Find  the  specific  gravity  of  a  bar  of  iron  5  in.  long,  and  2  in. 
square,  if  it  weigh  5  lbs. 

5  X  2  X  2  =  20  cu.  in. ;  if  20  cu.  in.  weigh  5  lbs.,  4  cu.  in.  weigh 
1  lb.,  and  1  cu.  in,  weighs  4  or.  1  cu.  in.  of  water  weighs 
^oz. 

_|-  =  6Hf  =  6.912.  Am. 

52.  Find  the  specific  gravity  of  a  bar  of  iron  18  in.  long,  2^  in. 
wide,  If  in.  thick,  if  it  weigh  18  lbs.  9  oz. 

18  X  2^  X  If  =  73^ ;  if  73^  cu.  in.  weigh  18  lbs.  9  oz.  or  297  oz., 

297 

1  cu.  in.  weighs  — - 

^     73^ 

II -^|f|  =  6mf  =  6.983.  Ans. 

53.  Find  the  number  of  cubic  inches  to  the  pound  of  iron,  if  its 
specific  gravity  be  7.48. 

21  fi 
1  oz.  of  water  =  —    cu.  in. 
125 

1  lb.  of  water  =  2162<i6  ^^^  -^ 
125 

1  lb.  of  iron  (specific  gravity  7.48)  =  ^^^^'^^  =  36B1  cu.  in.  Aru. 
^        ^  7.48x125       ^^ 

54.  Find  the  number  of  cubic  inches  in  2  lbs.  6J  oz.  of  gold,  if  its 
specific  gravity  be  19.36. 

216 
1  oz.  (av.)  of  water  =»  —  cu.  in. 
125 

1  lb.  (av.)  of  water  =  ^Hi^li^  ^^  -^ 


teachers'  edition.  261 


,  ,,     ,,       s    c   '  ^         216  X  16  X  5760 

1  lb.  (troy)  of  water  =      ,,,^^,,,      -.  m. 

,  „      ,,       .    f       .         216  X  16  X  5760  X  2if 
I  lbs.  (troy)  of  water  = 1,5^,000  ' 


2^1  lbs.  of  gold  (specific  gravity  19.36  or  19./^) 

^216x16x5760x2^ 

125  X  7000  X  192^^ 


=  316224  ^^  .^^  _  2.987  cu.  in.  Ans. 
105875 

55.  How  many  pounds  does  a  boy  lift  in  raising  a  cubic  foot  of 
stone  under  water,  if  its  specific  gravity  be  2|? 

Specific  gravity  of  stone  =  2J. 
Specific  gravity  of  water  =  1. 

Difference  =  H- 

1  cu.  ft.  of  water  weighs  62.5  lbs. 

1^  X  62.5  lbs.  =  93.75  lbs.  Ans. 

56.  A  square-built  scow  12  ft.  long,  6|-  ft.  wide,  sinks  5  in.  in 
water.  What  does  it  weigh,  and  how  many  pounds  will  be  required 
to  sink  it  7  in.  deeper? 

12  X  6^  X  T^  =  32^  cu.  ft.  6^  X  12  X  1  =  78 

62.5  78 

X  32J  X  62J 


2031. 2 J  lbs.  4875  lbs. 

=  1  t.  31 J  lbs.  (1)  Ans.  =  2  t.  8  cwt.  75  lbs. 

1  31} 

1  t.  8  cwt.  43|  lbs.  (2)  Ans. 

57.  A  square-built  scow  11  ft.  long,  5|-  ft.  wide,  weighs  320  lbs. 
and  is  loaded  with  750  lbs.  of  stone.  How  deep  will  it  sink  in  the 
water  ? 

11  X  5iXTV  =  4if  cu.ft. 
4^1  X  62.5  lbs.  =  300.78125  lbs. 
320  +  750  =  1070  lbs. 

3.557  in.  Ans. 
30078125)107000000.000 


262  ARITHMETIC. 


58.  How  many  tons  of  ice,  specific  gravity  0.93,  can  be  packed  in  a 
building  50  ft.  long,  40  ft.  wide,  20  ft.  high  ? 

50X40X20X62^X0.93^^^3^^^    ^^^^ 
2000  ^ 

59.  If  an  iceberg  weigh  0.9  of  an  equal  bulk  of  sea-wakr,  how 
many  cubic  yards  in  an  iceberg  40  rds.  long,  6  yds.  wide,  and  rising 
160  ft.  out  of  the  sea? 

40  rds.  =  660  ft. 
6  yds.  =    18  ft. 
660  X  18  X  160  =  1900800  cu.  ft.  =  70,400  cu.  yds. 

Now,  if  the  iceberg  weighs  0.9  of  the  weight  of  an  equal  bulk  of 
sea-water,  only  r^jj  of  the  iceberg  is  above  the  water. 
10  X  70,400  =  704,000  cu.  yds.  Ans. 

60.  If  a  cubic  foot  of  brick  wall  weigh  90  lbs.  and  contain  22 
bricks,  with  the  mortar,  what  is  the  weight  and  specific  gravity  of  a 
brick  and  its  share  of  mortar  ? 

90  -5-  22  =  ij\  lbs.  (1)  Ans. 
62J  ^  22  =  2||  lbs. 
4^  -H  2||  =  If  =  1.444.  (2)  Ans. 

61.  What  is  the  weight  of  a  brick  wall  40  ft.  long,  20  ft.  high,  and 
1  ft.  thick,  if  the  specific  gravity  of  a  brick  with  its  mortar  be  1.46; 
and  how  many  thousand  bricks  will  be  required  for  the  wall,  allow- 
ing 22  for  a  cubic  foot  ? 

40  X  20  X  =  1  800  cu.  ft. 
800  X  1.46  X  62.5  lbs.  =  73000  lbs. 

=  36^.  (1)  Ans. 
800x22  =  17,600.  (2)  Ans. 


Exercise  LXI 

1.   Reduce  24  gals,  to  liters. 

24  gals.  =  96  qta. 
96  X  0.946»  -  90.816'.  Am. 


2.   Reduce  10  lbs.  troy  to  kilo- 
grams. 

10  lbs.  =  120  oz. 
120  x31.104k  =  3732.480* 

^  Z.TSm^"'  -Ana. 


TEACHERS     EDITION. 


263 


3.  Reduce  50.5  cu.  yds.  to  cu. 
meters. 

50.5  X  0.765°^'"  =  SS.eS''^'".  Aiis. 

4.  Reduce  69jVV  ^i-  ^^  kilo- 
meters. 

69.17  X  1.609»''»  -  111.2945'^'«. 

5.  Reduce   12  A.    12  rds.   to 
hektars. 

12  sq.  rds.  =  0.075  A. 
12.075  X  0.405»>»  =  4.89ha.  Ans. 

6.  Reduce  10  cords  to  sters. 
10  X  3.624«t  =  36.24^'.  Ans. 

7.  Reduce  4  cwt.  24  lbs.  to 
kilograms. 

4  cwt.  24  lbs.  =  424  lbs. 

424  X  0.454'^K  =  192.496kg.  Ans. 

8.  Reduce   25  bu.   2  pks.   to 
hektoliters. 

25  bu.  2  pks.  =  816  qts. 
816  X  1.101  qts.  =  898.416> 

=  S.dSm^K  Ans. 


10. 

Reduce    to 

the 

common 

system 

3ha. 

2.471  A. 
X3 

7.413  A. 
X160 

66.08  sq. 

rds. 

X30^ 

2sq. 

yds 

nearly. 

7  A.  66 

)  sq.  rds.  2  sq 

yds 

nearly. 
Ans. 

11.   Reduce    to   the    common 
system  12.125ci>m. 

1.308  cu.  yds. 
X121 


15.8295  cu.  yds. 


X27 


23.3965  =  23.4  cu.  ft. 
15  cu.  yds.  23.4  cu.  ft.  Ans. 


9.   Reduce  to  the  common  sys- 
tem IS""*. 

0.621 
Xl5 

12.   Reduce    to  the    common 
system  101.251. 

1.0567  liquid  qts. 

xioii 

9.315  mi. 
X320 

106.9908 
=  107  liquid  qts.  nearly.  Ans. 

100.8  rds. 

0.908  dry  qts. 

X5i 

xioii 

4  yds.  nearly. 

91.935 

9  mi.  100  rds.  4  yds.  nearly. 

Ans. 

=  92  dry  qts.  nearly.  Ans. 

264 


ARITHMETIC. 


13.   Reduce    to   the    common 
system  20. 25". 

20.25"  =  20251 

1.0567  liquid  qts. 
X2025 


4)2139.8175  qts. 

535  gals,  nearly.  Am. 

0.908  dry  qts. 
X2025 

32)1838.7 

57  bu.  nearly.  Ans. 

14.   Reduce    to   the    common 
system  (troy  weight)  5^«. 

5kg  =  50008. 
15.432  grs. 
X5000 


24 

77160  grs. 

20 

3215  dwt. 

12 

160  oz.  15  dwt. 

13  lbs.  4  oz. 

13  lbs.  4  oz.  15  dwt.  Ans. 

15.   Reduce    to  the    common 

system  24»». 

0.276° 

X24 

6.624 

Xl28 

79.872 

=-  80  cu.  ft.  nearly. 

6  c.  80  cu.  ft.  nearly.  Ana. 


16.  Reduce    to  the    common 
system  62.5i". 

1.196  sq.  yds. 
X62| 

74.75  sq.  yds.  Ana. 

17.  Reduce  to  the  common  sys- 
tem (avoirdupois  weight)  1001^8. 

2.205  lbs. 
XlOOl 


2207.205  lbs. 


100 
20 


2207  lbs. 


22  cwt.  7  lbs. 


1  t.  2  cwt. 
1  t.  2  cwt.  7  lbs.  nearly.  Ana. 

18.  Find  in  acres,  etc.,  the 
area  of  a  field  if  its  length  be 
100°»  and  breadth  75°'. 

100  X  75  =  75001"'. 
1.196  sq.  yds. 
X7500 


160 


8970  sq.  yds. 
296sq.rd8.168q.yds. 
1  A.  136  sq.  rds. 

1  A.  136  sq.  rds.  16  sq.  yds.  Ana. 

19.  Determine  the  number  of 
cubic  meters  in  a  box  2  yds.  long, 
3  ft.  wide,  2^  ft.  deep. 

2  yds.  =  6  ft. 
6  X  3  X  2J  =  45  cu.  ft. 
=  1 J  cu.  yds. 
If  X  0.765«»>'»  =  1.275«'»>'».  Am. 


TEACHERS     EDITION. 


265 


20.   Determine  the  number  of 

0.9461 

cubic  yards  in  a  box  2""  long, 

Xl7 

75«"  wide,  50"=*  deep. 

16.0821 

2  X  0.75  X  0.50  =  0.75«^'». 

0.75  X  1.308  cu.  yds. 

16.082»^« 

=  0.981  cu.  yds.  Ans. 

»       X  2.205  lbs. 

35.461  lbs. 

21.   If  a  man  walk  75™  'a  min- 

X 1.841 

ute,  what  is  his  rate  in  miles  per 

65.284  lbs. 

hour? 

75  X  60  =  4500'"  =  4.5'^'^. 

65.284 

4.5  X  0.621  mi.  =  2.795  mi.  Ans. 

X$0.02i 

$1.47.  Ans. 

22.  If  cast-iron  weigh  7.113« 
per  cubic  centimeter,  how  many 
pounds  does  a  cubic  foot  weigh  ? 

As  the  weight  of  the  iron  is 

7.113^  per  cubic  centimeter, 

the  specific  gravity  is  7.113. 

7.113  X  62.5  lbs.  =  444.5625  lbs. 

Ans. 

23.  How  many  steps  2  ft.  6 
in.  long  will  a  man  take  in 
walking  a  kilometer  ? 

0.621  mi. 
X5280 

32788.8  ft. 

1311.5 


25)32788.8 
1311.5  steps  =  1312  steps.  Ans. 

24.  Find  the  value  of  a  car- 
boy (17  qts.)  of  sulphuric  acid, 
of  1.841  specific  gravity,  at  2^ 
cents  a  pound. 


25.  Find  the  value  of  a  car- 
boy (17|i)  of  nitric  acid,  of  1.451 
specific  gravity,  at  15  cents  a 
pound. 

2.205  lbs. 


38.588  lbs. 
X  1.451 


55.971  lbs. 
X  10.15 

$8.40.  Ans. 


26.  Find  the  weight  in  pounds 
and  in  kilograms  of  31 1  gals,  of 
the  best  alcohol,  specific  gravity 
0.792. 

31 1  gals.  -  124f  qts. 

0.9461 
Xl24f 

117.9351  of  water. 


266 


ARITHMETIC. 


X  0.792 


93.405"!?.  (1)  Ans. 

93.405 
2.205  lbs. 


205.958  lbs.  (2)  Ans. 

27.  If  the  specific  gravity  of 
Bea- water  be  1.026,  and  that  of 
olive-oil  be  0.915,  what  will  be 
the  weight  of  a  hektoliter  of  each 
in  pounds  and  in  kilograms? 

iw  =  1001  =  lOO^K. 

1.026 
X  100^8 

102.6''8.  (1)  Ans. 
X  2.205  lbs 


226.233  lbs.  (2)  Ans. 

0.915 
X  lOO^^K 


91.5''8.  (3)  Ans. 
X  2.205  lbs. 


201.76  lbs.  (4)  Ans. 

28.  Find  the  weight  in  pounds 
and  in  kilograms  of  the  air,  spe- 
cific gravity  0.00129206,  in  a 
room  7™  by  5",  and  3.5™  high, 

7  X  5  X  3J  =  122i'"»». 
122i«">»  of  water  =  122,500k«. 

0.00129206 
X  122500''« 

158.277JJ^«.  (1)  Ans. 


158.277 
X  2.205  lbs. 

349  lbs.  (2)  Ans. 


29.  Find  the  weight  in  pounds 
and  in  kilograms  of  the  air,  spe- 
cific gravity  0.00129206,  in  a 
room  23  ft.  long,  16  ft.  wide,  and 
10  ft.  high. 

23  X  16  X  10  =  3680  cu.  ft. 

3680 

X  62J  lbs. 

230000  lbs. 
X  0.00129206 


297.1738  lbs.  (1)  Ans. 

297.1738 
X  0.454kK 


134.9169''8.  (2)  Ans. 

30.  If  a  balloon  weigh  2^«, 
and  contain  10,000*  of  hydrogen 
gas,  specific  gravity  0.00008929, 
what  is  its  lifting  force  in  kilo- 
grams and  in  pounds  when  the 
air  has  a  specific  gravity  of 
0.00129206? 

0.00129206 

0.00008929 


0.00120277 

XlOOOO** 


12.0277^ 
2. 

10.0277''«.  (I)  Am. 


TEACHERS     EDITION. 


267 


10.0277 
X  2.205  lbs. 


22.111  lbs.  (2)  Ans. 


31.  If  a  pile  of  wood  be  1.2™ 
wide,  7"*  long,  and  2™  high,  how 
much  is  it  worth,  at  $4.50  a 
cord? 

1.2  X  7  X  2  =  16.8«t. 
16.8  X  0.276°  =  4.6368<'. 
4.6368  x|4^  =  $20.87.  Ans. 


32.   How  many  miles  will  be 
travelled  in   1   hr.    28   min.   21 
!.,  at  the  rate  of  50'^°'  an  hour? 


sec 


1  hr.  28  min.  21  sec.  =  1.4725  hr. 
50^^™  =  50  X  0.621  mi.  =  31.05  mi. 
1.4725  X  31.05  mi.  =  45.721  mi. 
Ans. 


33.  Find  the  time  of  travel- 
ling 31  mi.  180  yds.  at  1  min. 
25  sec.  per  kilometer. 

V^  =  0.621  mi. 
1  min.  25  sec.  =  1y\  min. 
31  mi.  180  yds.  =  31.1023  mi. 

31102.3  ^  621  =  50.08 
50.08  X  1^^  min.  =  70.95  min. 
=  1  hr.  11  min.,  nearly.  Ans. 


34.  What  is  the  weight  of  12 
cu.  yds.  16  cu.  ft.  720  cu.  in.  of 
earth  of  which  a  cubic  meter 
weighs  1 1.  17  cwt.  ? 


12  cu.  yds.  16  cu.  ft.  720  cu.  in. 
=  12.608  cu.  yds. 
icbm  _  1,308  cu.  yds. 
12608  ^  1308  =  9.639. 
1 1.  17  cwt.  =  37  cwt. 
9.639  X  37  cwt.  =  356.643  cwt. 
=  17  t,  16  cwt.  64  lbs.  Ans. 

35.  Find  the  weight  in  grams 
of  a  liter  of  mercury,  of  which  a 
cubic  inch  weighs  0.4925  of  a 
pound  avoirdupois, 

11  =  61.03  cu.  in. 
0.4925  lbs.  X  61.03  =  30.057  lbs. 
30.057  X  453.598  =  13633.55«. 

36.  How  many  yards  of  cloth, 
at  $3.12^^  a  meter,  should  be 
given  in  exchange  for  15*"  at 
$  2.75  a  yard  ? 

1  yd.  =  0.914°'. 
0.914  X  $  3^  =  $  2.856  per  yd. 
15  X  1.0936  yds.  =  16.404  yds. 

16.404  x$2|-  =  $45,111. 
45111  -J-  2856  =  15.79  yds.    Ans. 

37.  If  a  wine  merchant  buy 
3"  of  wine  for  1600  francs,  at 
what  rate.  United  States  money, 
does  he  pay  a  gallon,  reckoning 
25  francs  equal  to  $4.85? 

3"  =  3001. 
1600  francs  -^  300  =  5i  francs. 
$4,850  ^25  =  $0,194. 
$0,194x5^  =  $1,035  perl. 
11  =  1.0567  qts. 
=  0.264  gals. 
1035  ^264  =  $3.92.  Ans. 


268 


ARITHMETIC. 


38.  A  mill-wheel  is  turned  by 
a  stream  of  water  running  at  the 
rate  of  a  yard  per  second  in  a 
channel  5  ft.  wide  and  9  in.  deep. 
Determine  the  weight  of  water 
in  metric  tons,  supplied  in  12 
hrs.,  if  a  cubic  foot  of  water 
weigh  1000  oz. 

3x5x|=lUcu.ft. 
12  hrs.  =  43,200  sec. 


43200  X  Hi  cu.  ft. 

=  486000  cu.  ft. 
486000  X  1000  oz. 

=  486000000  oz. 

=  30375000  lbs. 
0.00045359  m.  t.  x  30375000 

=  13777.796  m.  t. 

Ans. 


Exercise  LXII. 


1.  Which  is  the  greater  ratio, 
6:8  or  6:9? 

5:8  =  f  =  M. 
.-.  6:9  is  greater. 

2.  Which  is  the  greater  ratio, 
7:  10  or  9: 12? 

7:10  =  ^  =  M. 
9:12  =  ^3.  =  |  =  M. 
.'.  9  :  12  is  greater. 


3.  Which  is  the  greater  ratio, 
8:  9  or  10: 12? 

8:9  =  f    =if. 
10:12  =  M  =  f  =  if. 
.'.  8  :  9  is  greater. 

4.  Which  is  the  greater  ratio, 
6: 12  or  8:  14? 

6:12  =  t%  =  ^  =  /t. 

8:14  =  ^. 

.".  8  :  14  is  greater. 


6.  Which  is  the  greater  ratio,  10  cwt. :  15  cwt.  or  $  7  :  |9? 

10  cwt. 


10  cwt. :  15  cwt.  = 


15  cwt. 


I  =  f- 


17 


|7:$9  =  p  =  i.        .-.  $7  :|9  is  greater. 


6.   Which  is  the  greater  ratio,  5  dys. :  7  dys.  or  8  ft. :  11  ft.? 
5dy8.:7dys.=|iZ!:»^  =  |^. 

8  ft. :  11  ft.  =  AiL  ^^=.^.       .-.  8  ft. :  11  ft.  is  greater. 
11  It. 


teachers'  edition.  269 


7. 

Which 
9  yds. 

is  the  greater  ratio,  9 

yds 

. :  6  yds. 

or  5  : 

3? 

5:3  = 

■i  =  ¥- 

.5:3  iE 

!  greater. 

8. 

Which 
fib.: 

is  the  greater  ratio,  1 

lb.: 

i  lb.  or 

fyd. 

:fyd.? 

f  yd.  :  I  yd.  =  f^  =  |.         .-.  f  yd. :  |  yd.  is  greater. 


Exercise  LXIII. 

1.   If  24  men  can  finish  some  work  in  14  days,  how  long  will  it 
take  21  men  to  do  it  ? 

21:24::14dys.  :what? 

8       2 
2^^^  dys.  =  16  dys.  Ans. 


2.   A  well  is  dug  in  13  days  of  9  hours  each.     How  many  days  of 
10  hours  each  would  it  have  taken  ? 

10:9::  ISdys. :  what? 

L><13  dys.  =  112  =  IIX  dys.  Ans. 
10       ^         10  ^^    ^ 


3.  A  man  who  steps  2  ft.  5  in.  takes  2480  steps  in  walking  a  cer- 
tain distance.  How  many  steps  of  2  ft.  7  in.  will  be  required  for  the 
same  distance  ? 

2  ft.  5  in.  =  29  in.  2  ft.  7  in.  -  31  in. 

31:29:  :  2480:  what? 

80 

^^^^^^^  =  2320.  2320  steps.  Ans. 

pX 


270  ARITHMETIC. 


4.   If  T^j  ton  cost  $6,  what  will  7f  cwt.  cost,  at  the  same  rate? 

.7fcwt.  =  ||t.  =  Ht. 
^:H::|6:what? 


^  ^^X5  75        ^ 

15 

5.   If  42  yds.  of  carpet  2  ft.  3  in.  wide  are  required  for  a  room,  how 
many  yards  2  ft.  4  in.  wide  will  be  required  ? 

2  ft.  3  in.  =  27  in.  2  ft.  4  in.  =  28  in. 

28:27::42yds. :  what? 

3 
2ZAl^  =  |yd8.  =  40iyds.  ^n.. 


6.  A  court  was  paved  with  950  stones,  each  If  sq.  ft.,  and  is  re- 
paved  with  836  stones  of  a  uniform  size.     Find  the  size  of  each. 

836:950:  :  If  sq.  ft.  :  what? 
25 

950X11  ^WxiI=2,Vsq.  ft.  ^n.. 
836  mxQ        ^     ^ 

2 

7.  If  a  train,  at  the  rate  of  ^j  of  a  mile  per  minute,  take  3^  hra. 
to  reach  a  station,  how  long  will  it  take  at  the  rate  of  f^  of  a  mile  a 
minute  ? 

1^5  :t«V: -SI  lira:  what? 

^  7      ;3      4      28       ^^ 

8.  If  a  post  4  ft.  8  in.  high  cast  a  shadow  7  ft.  3  in.  long,  how  long 
a  shadow  will  a  post  11  ft.  high  cast? 

4  ft.  8  in.  =  4f  ft.      7  ft.  3  in.  =  7J  ft. 
4J:ll::7ift.  :what? 

11  X71^_3x  11x29     ,^.f.       ikV,    ,,.        , 
~~W~ 14x1 ^      "  ^* 


teachers'  edition.  271 

9.  When  a  shadow  8  ft.  5  in.  long  i?  cast  by  a  post  5  ft.  7  in.  high, 
how  high  is  a  steeple  that  casts  a  shadow  of  211  ft.  at  the  same  time  ? 

8^^:211::5TVft.  :what? 

^^^  ^  ^tV  =  ;;^X  211x67  _  -^39  9 8   ^^  _  J39  ^^  ^.^5  j^. 
8t\  101  x;^  '^'  ^"^ 

10.  If  4  men  can  mow  a  certain  field  in  10  hrs.,  how  many  men 
will  it  take  to  mow  it  in  5  hrs.  ? 

2 
5  :  10  :  :  4  men  :  what?  ^        men  =  8  men.  Ans. 

11.  A  tap  discharging  4  gals,  a  minute  empties  a  cistern  in  3  hrs. 
How  long  will  it  take  a  tap  discharging  7  gals,  a  minute  to  empty  it  ? 

7  :  4  :  :  3  hrs. :  what  ?        ^-^  hrs.  =  If  hrs.  Ans. 
7  ^ 

12.  A  pipe  discharging  3  gals.  1  pt.  a  minute  fills  a  tub  in  4  min. 
20  sec.  How  long  will  it  take  a  pipe  discharging  83  qts.  a  minute 
to  fill  it? 

3  gals.  1  pt.  =  25  pts.       83  qts.  =  166  pts. 

4  min.  20  sec.  =  260  sec. 

130 

166  :  25  :  :  260  sec.  :  what  ?  ^^  ^  ^^^  =  39f |  sec.  Ans. 

83 

13.  If  both  pipes  of  Ex.  12  discharge  at  the  same  time  into  the 
tub,  how  long  will  it  take  to  fill  it? 

4-J-  min.  =  260  sec. 
95^  :  12| :  :  260  sec.  :  what  ? 
12^X260^   %       25     260 
95J  191      ^        1 

14.  How  long  will  it  take  to  fill  a  cistern  of  165  gals,  by  a  pipe 
that  fills  one  of  120  gals,  in  7  min.  16  sec.  ? 

16  sec.  =  j%  min. 

120:  165:  :  7/^  min.:  what? 

11 
165x^  ^  WXJ09  ^    „    ^     ^  9  ^.^  59^  ^^^  ^,^^_ 

120       120  x;^ 


272  ARITHMETIC. 


15.   A  ship  has  sailed  1800  mi,  m  a  fortnight.     How  long,  at  the 
same  rate,  will  it  take  for  a  voyage  of  5000  mi.  ? 

1800:  5000::  2  wks.:  what? 
25 


x2  =  50_5^^^g^ 


xm      9 

9 
5f  wks.  =  5  wks.  4  dys.  nearly.  Ans. 

16.  The  wheels  of  a  carriage  are  6  ft.  9  in.  and  9  ft.  6  in.  in  cir- 
cumference. How  many  times  will  the  larger  turn  while  the  smaller 
turns  3762  times  ? 

6  ft.  9  in.  =  6f  ft.        9  ft.  6  in.  =  9^  ft. 
9J:6|::  3762:  what? 

99 

6|_><3762  ^  ^X27xm^  ^  2673.  Ans. 

17.  If  ^\  of  a  ship  be  worth  $2167,  what  is  the  value  of  -^^  of  it? 

^^Vit^t::  $2167:  what? 

^  X  $2167  ^  25  X  7  X  2167  ^  $379225  ^  ^^^3^  ^g    ^^ 

18.  What  will  be  the  weight  of  18  cu.  ft.  432  cu.  in.  of  stone  of 
which  10  cu.  ft.  864  cu.  in.  weigh  14  cwt.  7  lbs.  ? 

10J:18J::  1407  lbs.:  what? 

67 

18^Xl407^;Zx73x.I^^7^4891^^^^  ^^^^ 

lOi  %lx^  2  ^ 

2        =  1  t.  4  cwt.  45J  lbs.  Am. 

19.  If  280  lbs.  of  flour  make  360  lbs.  of  bread,  how  many  four 
pound  loaves  can  be  made  from  1  cwt.  of  flour  ? 

280  :  100  ::  360  lbs.  :  what? 
9 

^^^]^^  =  ^  -  128f  lbs.         128i+  4  =  321.  Am. 


teachers'  edition.  273 

20.  If  a  column  of  mercury  27.93  in.  high  weigh  0.76  of  a  pound, 
what  will  be  the  weight  of  a  column  of  the  same  diameter  29.4  in. 
high? 

27.93  :  29.4  : :  0.76  lb. :  what  ? 

^i^  lb.  =0.8  lb.  ^m. 

0.19 

21.  How  many  francs  will  pay  a  bill  of  £100,  when  £42  10  s.  8d 
is  equivalent  to  1090.98  francs  ? 

£42  10s.  8^.  =  £42xV 

42y8^  :  100 :  :  1090.98  francs  :  what? 

1090^8X100 ,^^^^^ ^]^^l]m^m  =  2565  franc. 

42tV  m     m      1 

22-  What  will  be  the  weight  of  a  cube  whose  edge  is  2  ft.  2  in., 
when  a  cube  of  the  same  material  whose  edge  is  1  ft.  4  in.  weighs 
537.6  lbs.  ? 

2  ft.  2  in.  =  2^  ft.  1  ft.  4  in.  =  li  ft. 

(H)' :  {2i)^  -  537.6  lbs. :  what? 
It  : -¥tV  -  537.6  lbs. :  what? 
21 

m 

P     2m^m^mi lbs.  =  2306.85  lbs.  Ans. 
0      ^X^        10  20 

2         8 

23.  If  a  square  field  measuring  50  yds.  lOf  in.  on  each  side  be 
worth  |2710}f,  what  is  the  value  of  a  square  field  62  yds.  1  ft. 
each  way  ? 

50  yds.  lOf  in.  ==  50f  yds.        52  yds.  1  ft.  =  62^  yds. 
(50f )2 :  (62i)2  .  .  1 2710if  :  what  ? 
123904   34969  ..  |46080  .     j^^^.^ 
49     ■      9      ''''      17      -^  ^  ■ 
17  5 

_49_  ^3»Ex«-^  =  14165.  ^ns. 

m 


274  ARITHMETIC. 


24.  A  gains  4  yds.  on  B  in  running  30  yds.  How  much  will  he 
gain  while  B  is  running  97^  yds.  ? 

30:971:  :  4  yds. :  what? 
13     ;2 

^^^X^  yds.  =  13  yds.  Ans. 

25.  If  10  cu.  in.  of  gold  weigh  as  much  as  193  cu.  in.  of  water, 
what  is  the  size  of  a  nugget  weighing  as  much  as  a  cubic  foot  of 
water? 

193  :  1728  :  :  10  cu.  in. :  what? 
1728  X  10     17280 


193  193 


89||f  cu.  in.  Ans. 


26.  If  a  garrison  of  1500  men  have  provisions  for  13  months,  how 
long  will  the  provisions  last  if  it  be  increased  by  700  men  ? 

1500  +  700  =  2200. 

2200:  1500:  :  13  mo. :  what  ? 

15 
IM^<2^  mo.  =  W  mo.  =  8M  mo.  Ans. 
22 

27.  If  a  tree  38  ft.  high  be  represented  by  a  drawing  \\  in.  high, 
what,  on  the  same  scale,  will  represent  the  height  of  a  house  45  ft. 
high? 

38:45::l^in.  :what? 
||^in.  =  Win.  =  lf|in.  ^m. 

28.  If  a  country  630  mi.  long  be  represented  on  a  raised  map  by 
a  length  of  5J  ft.,  by  what  height  ought  a  mountain  of  15,750  ft.  be 
represented  on  the  map  ? 

630  mi.  =  3,326,400  ft.  h\  ft.  =  66  in. 

3326400  :  15750  : :  66  in. :  what? 
5 

mm 

16 


teachers'  edition.  275 

29.  A  train  travels  ^  of  a  mile  in  18  sec.  How  many  miles  an 
hour  does  it  travel  ? 

1  hr.  =  3600  sec.        18  :  3600  :  :  ^  mi. :  what  ? 
200 

M2^^  =  50mi.^ns. 

30.  If  4J  tons  of  coal  fill  a  bin  9  ft.  long,  5  ft.  broad,  5  ft.  high, 
how  many  cubic  feet  will  be  required  for  the  coal  of  a  steamer  carry- 
ing 3  weeks'  consumption  at  20  tons  a  day  ? 

9  X  5  X  5  =  225  cu.  ft.        3  wks.  =  21  dys. 

21  X  20  t.  =  420  t.  4^  :  420  : :  225  cu.  ft. :  what  ? 

25 
420XMX_2 eu  ft.  =  21.000  cu.  ft.  ^ns. 

31.  If  2  lbs.  of  rosin  be  melted  with  5  oz.  of  mutton  tallow,  to 
make  a  grafting* wax,  how  many  ounces  of  tallow  will  20  oz.  of  the 
wax  contain  ? 

2  lbs.  +  5  oz.  =  2  lbs.  5  oz.  =  37  oz. 

37  :  20  :  :  5  oz.  :  what  ? 

20x5      100  o2fi  A 

— ^^^-  = oz.  =  2  If  oz.  Ans. 

37  37  '^ 


Exercise  LXIV. 

1.   How  many  days  8  hours  long  will  60  men  take  to  finish  some 
work  which  24  men  can  do  in  15  days,  working  10  hours  a  day  ? 

^1  ^^::  15  dys. -.what? 
60  I  24  ^ 

5       3 

^  X  ^P 
2 


276 


ARITHMETIC. 


2.  What  will  be  the  expense  of  covering  a  roorn  with  drugget  4  ft. 
wide,  at  91 1  cts.  a  yard,  when  carpet  2  ft.  3  in.  wide  for  the  room 
costs  $  70.50.  at  $  1 .37^  a  yard  ? 

$0.91|  =  |H-        $1.37i  =  $lf. 
4    2i 
If    H 

Lllxlilxlxi  =  ^^  =  $26.44.^n,. 


$70}:  what? 


n 

4 


i      11 


16 


3.  If  4418  tons  of  iron  ore  produce  $36,190  worth  of  metal,  when 
iron  is  at  $37.50  a  ton,  what  will  be  the  value  of  the  iron  from  2275 
tons  of  ore,  at  $47  a  ton  ? 

37}  I  47 
4418  I  2275 


$36,190:  what? 


91  335 

2  X  ^T  X  m^  X  mX9^  ^  $70070  ^  ^23.356.67   Ans 

n  X  im  3 

3      H 

4.  If  a  bar  of  iron  3^  ft.  long.  3  in.  wide,  2|  in.  thick  weigh  93 
lbs.,  what  will  be  the  weight  of  a  bar  3f  ft.  long,  4  in.  wide,  and  2^ 
in.  thick  ? 


93  1bs. :  what? 


31 
3 

3f 

4: 

2| 

2i 

31 

xfx^x|-X:^  =  124  lbs.  Atu. 
^        I       1)3      Jl 


6.  If  40  bu.  of  wheat  can  be  grown  on  the  same  area  as  48  bu.  of 
barley,  and  28  acres  produce  840  bu.  of  wheat,  how  much  barley  will 
be  obtained  from  38  acres? 


40 


840bu. :  what? 


6 
6  30 

^><^8iiM!»i368bu. 


TEACHERS     EDITION. 


277 


6.  If  18  men  can  dig  a  trench  150  ft.  long,  6  ft.  broad,  and  4  ft. 
6  in.  deep  in  12  days,  how  long  will  16  men  take  for  a  trench  210  ft. 
long,  5  ft.  broad,  and  4  ft.  deep  ? 


12dys. :  what? 


i      3^ 


16 

18 

150 

210 

6 

5 

4i 

4 

^ 

7 

7.  In  the  reprint  of  a  book  consisting  of  810  pages,  50  lines  are 
contained  in  a  page,  instead  of  40,  and  72  letters  in  a  line,  instead  of 
60.     Of  how  many  pages  will  the  new  edition  consist  ? 


810:  what? 


10      6       J 

^0  X  ^0  X  ^ 

mxn 


=  540.  Ans. 


8.   If  3280  42-lb.  shot  cost  $3000,  how  many  32-lb.  shot  can  be 
bought  for  $4200? 


3000  I  4200 
32    42 


3280:  what? 


7        21 

^m  xnx 
?>m  X  n 


41 


6027. 


9.   "What  must  be  the  rate  of  wages,  that  12  men  may  earn  in  10 
days  the  same  amount  that  9  men  earn  in  14  days,  at  $  1.50  a  day  ? 
12  19 
10  I  14 


:  $1.50:  what? 


0.05 

7        O.I^ 

Qx^^xW^  =  $M5^cpi.575.  Ans. 
1%XX^  2 

2 


278 


ARITHMETIC. 


10.  A  reservoir  15  yds.  long  and  4  ft.  deep  holds  32,500  gals. 
Determine  the  quantity  of  water  it  will  hold  when  it  has  been 
increased  in  length  by  18  ft.  and  in  depth  1  ft. 


15    21 
4 
7  8125 


::  32,500  gals. :  what? 


;^x^ 


gals.  =  56,875  gals.  Ans. 


11.  How  far  can  A,  who  takes  3.1  ft.  each  step,  run,  while  B, 
who  takes  2.3  ft.  each  step,  runs  220  yds.,  if  A  takes  7  steps  while 
Stakes  11? 


220yd8. :  what' 


2.3 

^•^••22( 

11 

7    '-^^^ 

20 

3.1  X  7  X  m 

2 

.3x;; 

yds. 


434 
2.3 


yds.  =  188^1  yds.  Am. 


12.  If  6  hours  be  required  for  travelling  a  given  distance  at  a 
given  rate,  how  long  will  be  required  when  the  distance  is  diminished 
by  one-fourth  and  the  rate  is  increased  by  one-half? 


6  hrs. :  what  ? 


X  ^  hrs. 


3  hrs.  Ans. 


13.  IIow  many  hours  a  day  must  5  men  work  to  mow  the  same 
quantity  of  grass  in  8  days  that  7  men  can  mow  in  6  days,  working 
10  hours  a  day  ? 

3      g 

10  hrs. :  what  ?        '^^Xfx^^  hrs.  =  ^  hre.  -  lOi  hrs. 
?X? 
i 
2 


6 


14.  If  a  bar  10  ft.  6}  in.  long,  3f  in.  broad,  3J  in.  thick  weigh  4 
cwt.  8.23  lbs.,  what  length  must  be  taken  to  weigh  a  long  ton  when 
the  breadth  and  thickness  are  4f  in.  and  4J  in.  respectively? 


TEACHERS     EDITION. 


279 


1  1.  t.  = 

-  2240  lbs. 

40823 

224000 

4f 

3f          :: 

7000 

xm0 

4  cwt.  8.23  lbs.  =  408.23 


lOMft. :  what? 


23 


40823  X  ;t^  X  33  X  ^  X  ^  X  ^^     ■  ^^^^^  • 

2      XX  3 


38  ft.  1.5  in. 


15.  If  27  men,  in  28  days  of  10  hours  each,  dig  a  trench  126  yds. 
long,  2^  yds.  broad,  1}  yds.  deep,  how  long  a  trench  2|  yds.  broad, 
If  yds.  deep,  will  56  men  dig  in  25  days  of  S^  hours  ? 


27 

56 

10 
28 

8i 

25:  :126  yds.:  what? 

2| 
If 

2J 

1^                                             6 

%      %       ^                           n 

^  X  ^  X  ^^  X  33  X  25  X  ^  X  3  X  ;  ^^  „^  „ 

XI 

x7x^Jx;px^x^^x^x;2  '^^' 

150  yds.  An%. 


16.  What  must  be  the  length  of  a  bar  of  silver  f  in.  square,  that 
it  may  weigh  the  same  as  a  bar  of  gold  \  in.  square  and  6|  in.  lonpr, 
if  the  weight  of  a  cubic  inch  of  silver  have  to  that  of  a  cubic  inch  of 
gold  the  ratio  47  :  88  ? 


(f )^  i  (^)'  :  :  6|  in,  :  what ' 

47    88         ^ 


47 


what? 


4       22       3 

i^L><Mxi^in.  =  5|fin.^ns. 
^  X  47  X  ^  X  ;f  ^^ 

17.    If  it  take  34^k  of  wool  to  make  25«  of  cloth  0.6"*  wide,  how 
long  a  piece  of  cloth  0.8™  wide  can  be  made  from  108.8''«  of  wool  ? 

4 

xn 

34    108.8  .,^^,  ^^^,         W-^X  0.6X25  ^  ^^^   ^^^ 
0.8        0.6  3^X0.^ 


280  ARITHMETIC. 


18.  An  oak  beam  5.40"  long,  0.63"  thick,  and  0.57"  wide  weigh 
1469.25''8 ;  find  the  weight  of  a  beam  whose  dimensions  are  4.87° 
0.58"  0.53". 


4.87 

0.58::  1469^8:  what? 

0.53 


5.4 
0.63 
0.57 

29  653 

487  X  ^'X  53  X  mf"^     488782907*8 


^^0X63X57X4  430920     =  1134.2776K  ^n.. 

m 

30 

19.  A  certain  quantity  of  air  has  a  volume  of  195.5  cu.  ft.  at  27.8°. 
What  will  be  its  volume  at  100°  ? 

100°  -  27.8°  =  72.2°.  72.2  x  0.00367  =  0.264974. 

1  :  1.264974  ::195Jcu.  ft.:  what? 

1.264974  xl95i        -,       „._„        .,     , 
-— ^ *  cu.  ft.  =  247.3  cu.  ft.  An&. 

20.  A  quantity  of  air  at  a  temperature  of  15.6°  C.  has  a  volume 
of  4  cu.  ft.  under  a  pressure  of  12  lbs.  to  the  square  inch.  What  will 
be  its  volume  at  a  temperature  of  48.7°  C,  and  under  a  pressure  of 
14  lbs.  the  square  inch  ? 

48.7°  -  15.6°  =  33.1°.  33.1  x  0.00367  =  0.121477. 

't|u21477  =  ^^^"-^^-^^^^^ 
3  % 

;;Zx  1121477  x^^„  ff      3364431  _  c,      ^c,        ,,     . 

7    mm 

125000 


Exercise   LXV. 

1.   Divide  $12,000  proportion- 
ally to  the  numbers  3,  4,  5. 
3  +  4+5  =  12. 

1000 
^lxf^-»  =  14000. 

1000 
3      11^000^^30^ 

1000 
/j^X^'^f^- 15000. 

TEACHERS     EDITION. 


281 


2.    Divide  815  tons  proportion- 
ally to  I  I  I  f 

ih  I  I  f )  X  60  =  30,  40,  45,  48. 
30  +  40  +  45  +  48=163. 
5 

150  t. 


30      8^ 
X0      1 


5 

:^  =  200t. 

i8.W  =  240t. 


_40 
X0 

45 


;^3 


3.  Divide  6853  lbs.  of  wool 
proportionally  to  If,  2f ,  5f  ;  and 
also  proportionally  to  the  recipro- 
cals of  these  numbers. 

ih  -¥.  ¥)  X  60  =  105,  168,  350. 
105  +  168  +  350  =  623. 
11 


105  m^ 
m     1 

11 

168  ^  m^ 


m 


1155  lbs. 


1848  lbs. 


3850  lbs. 


350  0§f3 

m     1 

The  reciprocals  of 

li2f,5t  =  f  T\./3• 
(f.A,A)x70=40,25.12. 
40  +  25  +  12  =  77. 


89 

40^2^^  =  3560  lbs. 
JJ         1 

89 
25xW?  =  2225  1bs. 

n     1 

12  X  si  =  1068  lbs. 

n     1 


4.  Two  persons  join  in  pur- 
chasing some  property,  one  pay- 
ing §1250,  and  the  other  |1000. 
If  the  property  rise  in  value  to 
$3600,  what  will  be  the  value  of 
each  one's  share  ? 


1 1000 +  $1250 
200  8 


$2250. 


%m     1 

250  8 


$1600. 


$2000. 


5.  Gun-metal  is  composed  of 
3  parts  (by  weight)  of  tin  to  100 
parts  of  copper.  What  weight 
of  each  of  these  metals  will  there 
be  in  cannon  weighing  721  lbs.  ? 

3  +  100  =  103. 
7 
—  X^  =  21  lbs.  tin. 

in    1  ' 


100,  /7;^;      ^^^1, 

—  X-^=/001bs.cop. 


282  ARITHMETIC. 


6.  Bell-metal  contains  78  parts  copper  and  22  parts  tin.  What 
weij^ht  of  each  of  these  metals  will  there  be  in  a  bell  weighing 
937  lbs.? 


.  78  +  22  = 

=  100. 

TV\rX937  = 

=  730.86  lbs. 

copper. 

t^  X  937  = 

=  206.14  lbs. 

tin. 

7.  It  takes  75^«  of  saltpetre,  12.5i^8  of  charcoal,  and  12.5*8  of 
sulphur  to  make  100''«  of  powder.  How  much  of  each  of  these 
substances  will  be  required  to  make  10,000,000  cartridges,  each  con- 
taining 58  of  powder  ? 


75  + 12| 

+  12^  =  105. 

10,000,000x58  = 

=  50,000,0008  =  50,000^8. 

500 

1 

=  37,5001^8  saltpetre. 

125 

xm 

50 

1 

=  6250''8/^^^^«°^l- 
I  sulphur. 

8.  Yellow  copper  contains  2  parts  of  red  copper  and  1  part  zinc. 
How  many  ounces  of  red  copper  are  there  in  an  article  weighing  1  lb. 
made  of  yellow  copper  ? 

1  lb.  =  16  oz. 

2  +  1  =  3. 

I  X  Ji&  -  ^  =  10|  oz.  Am. 

9.  Type-metal  is  made  of  an  alloy  containing  39  parts  of  lead  to 
11  parts  antimony.  How  many  pounds  of  each  will  be  required  to 
make  957  lbs.  of  type  ? 

39  +  11=50. 
*!  X  ^P  =  HV  =  746.46  lbs.  lead. 
1^  X  »F  =  ^^  =  210.54  lbs.  antimony. 


teacher's  edition.  283 

10.  Plumber's  solder  contains  2  parts  lead  and  1  part  tin.  How 
much  of  each  of  these  in  100  lbs.  of  solder  ? 

2  +  1  =  3.  f  X  100  =  66f  lbs.  lead.  ^  X  100  =  33i  lbs.  tin. 

11.  The  air  is  composed  of  oxygen  and  nitrogen.  In  100  volumes 
of  air  there  are  21  volumes  of  oxygen  and  79  of  nitrogen.  Reckon- 
ing the  weight  of  a  liter  of  oxygen  to  be  1.4295^,  that  of  a  liter  of 
nitrogen  1.2577«,  find  the  number  of  grams  of  each  gas  in  lOOs  of  air. 

21  X  1.4295  =  30.0195. 
79  X  1.2577  =  99.3583. 
30.0195  +  99.3583=129.3778. 

300195  of  100«  =  ^019500_«^  23.203^0. 


1293778  1293778 

1008  -  23.203«  =  76.797s  H. 

12.  What  is  the  value  of  the  gold  in  a  chain  weighing  3  oz.  4  dwt., 
supposing  it  to  be  18  carats  fine  (that  is,  18  parts  of  pure  gold  out  of 
24),  at  $  19  an  ounce  ? 

3  oz.  4  dwt.  =  3^  oz. 
3^X|19  =  |60f. 
i|of$60f  =  $45.60.  Ans. 

Exercise  LXVI. 

1.  Arnold  and  Baker  enter  into  partnership.  Arnold  puts  in 
$6000  for  8  months,  and  Baker  $4000  for  6  months.  Their  profits 
are  $  2000.    What  is  each  man's  share  ? 

8  X  $6000  =  $48,000. 
6  X  $4000  =  $24,000. 
48,000  +  24,000  =  72,000. 

|^2P0     $2000^^^333  33^.^^ 

3 

g»     $2000^^333  37  g,^^ 

WPP       1 

3 


284  ARITHMETIC, 


2.  Dobflon  furnishes  the  firm  of  Dobson  &  Fogg  with  |5000  for  13 
months;  Fogg  furnishes  $7000  for  9  months.  Their  profits  are 
$  1700.     What  is  the  share  of  each  ? 

13  X  1 5000  =  $  65,000.  9  x  $  7000  =  $  63,000. 

65,000  +  63,000  =  128,000. 

65  425 

mm  X  liW  =  117625  ^  ^  gg3  28.  Dobson's. 
X^^m         1  32 

x^ 

32 

1 1700 -$863.28  =  $836.72.  Fogg's. 

3.  In  a  business  speculation,  A  furnishes  $800,  and  after  3  months 
$250  more ;  B  furnishes  $950,  and  at  the  end  of  2  months  withdraws 
$200 ;  C  furnishes  $650,  and  at  the  end  of  6  months  $400  more.  At 
the  end  of  a  year  they  realize  a  profit  of  $2516.  How  shall  it  be 
divided  among  them  ? 

A.  B.  C. 

12  X  $800  =  $9600  12  X  $950  =  $11400  12  x  $650  =  $7800 

9x    250=    2250  10  x    200=     2000  6x    400=    2400 

$11850  $9400  $10200 

$11,850  +  $9400  +  $10,200  =  $31,450. 
237  4  188  4 

Xm^  X  ^^^^^  -  $948    A's  ^^^^  X  ^^  -  $752   B'a 

m  m 

204  4 

»ix^^  =  $816,  C'8. 


4.   Two  partners,  A  and  B,  begin  business  with  capitals  of  $3500 
and  $8700,  and  A  is  to  have  0.12  of  the  profitw  for  managing  the 
business.     How  shall  a  profit  of  $1906.25  be  divided  between  them  ? 
0.12  of  $  1906.25  =  $  228.75.  $  1906.25  -  $  228.75  -=  $  1677.50. 

3500  +  8700  =  12200. 
35 

^Xif5-f481.25. 

122 

$481.25 +  $228.75  =  $710,  A'b. 
$1906.25 -$710  =  $1196.25,  B's. 


teachers'  edition.  285 

5.  A  puts  $2100  into  a  business,  and  B  $1750.  At  the  end  of  a 
year  each  puts  in  $700  more,  and  C  joins  them  with  $2500.  At  the 
end  of  18  months  from  this  time  how  shall  a  profit  of  $2166.50  be 
divided  ? 

A.  B.  C. 

30  X  $2100  =  $63,000  30  x  $1750  =  $52,500  18  X  $2500  =  $45,000. 
18  X   700=  12,600  18  X   700=  12,600 

$75,600  $65,100 

7560  +  65,100  +  45,000  =  185,700. 

63 

W  217 

m       14  m        7 

mM  ^  mm0  ^  $882.  a's.    mm  x  mmm  ^  ^^,,^,0,  b's. 

3 

m       175 


m^ 


6.  Three  graziers  hire  a  pasture,  for  which  they  pay  $132.50. 
One  puts  in  10  oxen  for  3  months,  another  12  oxen  for  4  months,  and 
the  third  14  oxen  for  2  months.  How  much  of  the  rent  ought  each 
to  pay  ? 

3  X  10  =  30  5 

4x12  =  48  12         m 

2  X  14  =  28  i^  X  ^^^^  =  $60.  (2) 

106  ^ 

5 
15         1%^  5 

m  "^  "w"  "  ^'^•'''  ^'^  m,  X  ^^^ = $35.  (3) 

2  i 


286  ARITHMETIC. 


7.  A  begins  business,  with  a  capital  of  $  2400,  on  the  19th  of 
March  ;  and  on  the  17th  of  July  admits  B  as  a  partner,  with  a  capital 
of  $  1800.     Dec.  31  the  profits  are  $  943.     What  is  the  share  of  each  ? 

From  March  19  to  Dec.  31  is  288  dys. 
From  July  17  to  Dec.  31  is  168  dys. 

288x12400  =  $691,200 

168  X     1800=    302,500 

$993,600 
16  41 

?W^^X^^  =  $656.A'8. 

mm     1 

wWxM?  =  $287.B'8. 

8.  A  and  B  join  capitals  in  the  ratio  7 :  11.  At  the  end  of  7 
months  A  withdraws  \  of  his,  and  B  |  of  his  ;  and,  after  11  months 
more,  they  divide  a  profit  of  $5148.50.     What  is  the  share  of  each  ? 

18x7   =126  18x11  =  198 

11X3^  =  J8^  llx3j=    40^ 

87^  =  ^.  157f  =  ^. 

525  +  946  =  1471. 
175 

21       %m 


2 
$  5148.50  -  $  1837.50  =  $  331 1,  B's. 

9.  Divide  £65  9  8.  among  three  persons,  ho  that  the  first  may  liave 
as  many  half-crowns  as  the  second  has  shillings;  and  the  second  as 
many  guineas  as  the  third  has  pounds. 

Ist  has  2\  times  as  much  as  2d. 
2d  has  1^  as  much  as  3d. 


teachers'  edition.  287 

3d   has      1  part.  7 

2d    has    fipart.  1^  x  i*  =  735  s.  =  £36  15  s. 

1st  has    -V-part.  W        1 

7 
3d   has    40parts.  i^  x  i»  -  294s  -  £14  14. 

2d    has    42  parts.  —  X -^  -  ^y4s.  -  £  14  i4s. 

1st  has  105  parts.  w 

All  have  187  parts.  —  x  ^W=  280  s.  =  £14. 

£65  9s.  =  1309s.  W        1 

10.  Two  partners  begin  business  each  with  a  capital  of  |2000. 
A  adds  $  500  at  the  end  of  2  months,  and  $  500  at  the  end  of  7 
months  ;  B  adds  $  800  at  the  end  of  3  months.  What  is  the  share  of 
each,  at  the  year's  end,  of  a  profit  of  $  3605.25  ? 

12  x  1 2000  =  $  24000  12  x  $  2000  =  1 24000 

10  X      500=      5000  9x       800=      7200 

5  x      500  =      2500 


131200 


$  31500 
31,500  +  31,200  =  62,700. 

21 

n^  345 

^»  xi»^^  =  1 1811.25,  A'8. 

nun     w 

4 
I  3605.25  -  $  1811.25  =  $  1794,  B's. 


Exercise  LXVII. 


1.   The  population  of  a  town  in  1870  was  12,275,  and  it  increased 
1q  in  the  next  ten  years.     Find  its  population  in  1880. 

If  100  represent  the  population  in  1870,  then  108  will  represent 
the  population  in  1880. 
27         491 
»  of  iM£  =  13,257.^.3. 

im      1 
it 


288  ARITHMETIC. 


2.  How  much  metal  will  be  obtained  from  365  tons  of  ore,  if  the 
metal  be  7%  of  the  ore  ? 

If  100  represent  the  ore,  then  7  will  represent  the  metal  in  the  ore. 

73 

J_  of  ?^  =  ^  =  25.55  tons.  Am. 
l^        1        20 
20 

3.  If  gunpowder  contains  75%  of  saltpetre,  10%  of  sulphur,  15%  of 
charcoal,  how  much  of  each  is  there  in  a  ton  of  gunpowder  ? 

If  100  represent  the  gunpowder,  then  75,  10,  and  15  will  repre- 
sent respectively  the  saltpetre,  sulphur,  and  charcoal. 

20 
^     m9^  1500  lbs.  saltpetre. 

m     1 

20 
10  ^»^^  200  lbs.  sulphur. 

m     1 

20 

1|^X^  =  300  lbs.  charcoal. 
%^9        1 

4.  A  manufactory  uses  24  tons  of  coal  a  day,  and  20%  of  it  is  lost 
in  smoke.  How  much  coal  would  be  needed  if  this  waste  could  be 
prevented  ? 

If  100  represent  the  number  of  tons  of  coal  used,  then  20  will 
represent  the  number  of  tons  lost  in  smoke. 

^X^  =  ^  =  4.8t.    24t.-4.8t.  =  19.2t.  Am. 
J.^      1       5 
5 

5.  Air  consists  of  20.0265%  (by  measure)  of  oxygen  gas  and 
79.9735%  of  nitrogen.     How  much  oxygen  in  1750  cu.  ft.  of  air? 

If  100  represent  the  number  of  cubic  feet  of  air,  then  20.0265 

will  represent  the  number  of  cubic  feet  of  oxygen. 

35 

20.0265  ^;W  oKt^Aa        f.     a 
—    X  ^^-~  =-  350.46  cu.  ft.  Am. 


teachers'  edition.  289 

6.   A  town,  after  decreasing  25%,  lias  4539  inhabitants.     Find  its 
number  at  first. 

If  100  represent  the  population  at  first,  then  75  will  represent 
the  .population  now. 
4       1513 


7.  2%  of  a  regiment  of  750  men  are  killed  in  an  engagement,  6% 
are  wounded,  and  4%  are  missing.  What  is  the  number  still  availa- 
ble for  service  ? 

If  100  represent  the  number  of  men  in  the  regiment  then  2,  6, 

and  4  will  represent  the  number,  killed,  wounded,  and  missing. 

22        30 

2  +  4  +  6  =  12  W  ^JW      aan  a 

22L  X  -^-^  =  660  men.  Ans. 

100  -  12  =  88  X0^       1 

8.  If  3|  tons  of  sulphur  are  required  to  make  31i  tons  of  gunpow- 
der, what  is  the  per  cent  of  sulphur  in  gunpowder  ? 

If  100  represent  the  whole  weight,  then  the  number  required  to 

33 
represent  3|  tons  of  sulphur  will  be  — ^  of  100  =  12. 

That  is,  12%.  Ans, 

9.  In  a  school  of  80  children,  17|%  are  girls.  Find  the  number 
of  boys. 

If  100  represent  the  number  of  scholars  in  the  school,  then 
100  —  17^,  or  82|,  will  represent  the  number  of  boys  in  the 
school. 

2 
33  ^ 

^-of80  =  ^X  — X^  =  66boys.  Ans. 
100  ^       X9^      I 

10.  If  goods  are  bought  for  $415,  and  sold  for  |500,  what  is  the 
gain  per  cent? 

1 500  -  1 415  =  1 85,  actual  gain. 


290  AlilTHMETIC. 


Since  the  gain  on  |415  is  $85,  the  gain  on  100  is 
17 
1^  of  ^  ^  20t§ %.     .-.  the  gain  is  20tf  %  Ans. 

W      1 
83 

11.  If  goods  are  bought  for  $415,  and  sold  for  $400,  what  is  the 
loss  per  cent  ? 

$415  -  $400  =  $15,  actual  loss. 
Since  the  loss  on  $415  is  $15,  the  loss  on  100  is 
3 

1^  of  ^  =  m     .'.  the  loss  is  3H %.  Ans. 

83 

12.  $  500  is  4  %  of  what  number  ? 

If  4  represent  $500,  100  will  represent  ^^  of  $500  =  $  12,500. 

13.  A  farmer  buys  24  head  of  cattle  at  $80  a  head,  and,  after 
losing  6,  sells  the  remainder  at  $  105  a  head.  How  much  does  he 
gain  or  lose  per  cent  ? 

24  head  of  cattle  at  $80  per  head  cost  $1920. 
18  head  of  cattle  at  $  105  per  head  cost  $  1890. 
$  1920  -  $  1890  =  $  30,  actual  loss. 
Since  the  loss  on  $1920  is  $30,  the  loss  on  100  is 
25 
^  of  ^  =  \^^.     .-.  he  loses  l^^%.  Ans. 

H 
16 

14.  If  a  ton  (2240  lbs.)  of  ore  in  a  gold  mine  yield  5  oz.  (troy)  of 
gold,  what  is  the  yield  per  cent? 

5  oz.  troy  =  ^j  lbs.  troy  =  t*^  of  ^^§g  lbs.  av.  =  H  lb.  av. 
If  100  bo  taken  to  represent  the  ore,  the  number  required  to 
represent  the  metal  will  be 

3     ;z0 

22^^^^^^-^0><i><f -lie- ^^^^^^-^^^^^-/o-^- 
m    7 

28 


teachers'  edition.  291 

15.   If  the  ore  in  a  mine  yields  -^^  of  1%  of  pure  gold,  how  many- 
tons  (2240  lbs.)  of  ore  must  be  taken  to  obtain  7  lbs.  (troy)  of  gold? 
7  lbs.  troy  =  7  X  -fHt  l^s.  av.  =  5if  lbs.  av. 
If  100  be  taken  to  represent  the  ore,  -^^  will  represent  the  gold 
in  the  ore.     If  ^\  represent  5||  lbs.,  100  will  represent 

1^  of  5if  ^  48 

lbs  or.M :!tons-Wx^X^-^X-^ 

=  -4/  =  6f  t.  Ans.       7 


S-^0 


16.  12J  tons  of  iron  are  obtained  from  235  tons  of  ore.  What 
per  cent  of  the  ore  is  iron  ? 

If  100  be  taken  to  represent  the  whole  weight  of  the  ore,  the 
number  required  to  represent  12^-  tons  will  be 

50 
Ifi  of  100  =  ?^  X  -i-  X  »  =  5if .     That  is,  b\%.  Ans. 
47 

17.  Goods  are  sold,  at  a  loss  of  37o,  for  $2667.50.  What  was  the 
cost? 

If  100  be  taken  to  represent  the  cost,  100  —  3  =  97  will  represent 
the  selling  price.  Therefore  the  selling  price  was  ^-^^  of 
$2667.50  =  12750.  Ans. 

18.  Teas  at  68  cents,  86  cents,  and  96  cents  a  pound,  are  mixed 
in  equal  quantities,  and  sold  at  90  cents  a  pound.  Find  the  gain 
per  cent. 

68  +  86  +  96  _  g3i  ^^^^  ^^^  ^^^     g^  _  ggi  _  g2_  ^^^^^^  ^^^^^ 


3 

100 
Since  the  gain  on  83^  cents  is  6|  cents,  the  gain  on  100  is  -— • 

of  6f  =  8.     That  is,  8%.  Ans.  ^^^ 

19.   By  selling  goods  for   $1173.92,  a  merchant  gains  $153.12. 
Find  the  gain  per  cent  on  the  cost. 

$  1173.92  -  $  153.12  =  $  1020.80,  cost. 

Since   the  gain   on   $1020.80   is   $153.12,  the  gain  on  100  is 
100 


1020.80 


of  153.12  =  15.     That  is,  15  %.  Ans. 


292  ARITHMETIC. 


20.   If  to  25  gals,  of  alcohol  2  gals,  of  water  are  added,  how  much 
per  cent  of  the  mixture  is  w'ater?  how  much  per  cent  is  alcohol? 

25  +  2  =  27  gals.,  the  whole  amount. 

If  100  be  taken  to  represent  the  whole  amount,  the  number 

required  to  represent  2  gals,  will  be  ^^  of  100  =  7^f ,     That 

is,  7^  J  %  water.  Ans. 
100  _  7^^  =  92^f  %  alcohol.  Am. 


21.  "What  was  the  cost  when  17^%  was  gained  by  selling  goods 
for  $253.80? 

If  100  be  taken  to  represent  the  cost,  100  +  17^  =  117^  will 
represent  the  selling  price. 

Therefore,  the  cost  was  -^  of  $  253.80  =  $  216.  Am. 
1. 1.  (-^ 


22.  A  wine  merchant  mixes  24  gallons,  at  $7  a  gallon,  with  18 
gallons,  at  $5  a  gallon,  and  sells  the  whole  at  $7  a  gallon.  What 
does  he  gain  per  cent  ? 

24  X  |7  +  18  X  $5  =  $168 +  $90  =  1258,  cost. 
24  +  18  =  42,  whole  number  of  gallons. 
42  X  $  7  =  $  294,  selling  price, 
$  294  -  $  258  =  $  36,  actual  gain. 

Since  the  gain  on  $  258  is  $  36,  the  gain  on  100  is  \%%  of  36  =  13f|. 
That  is,  13||%.  Am. 

23.  By  selling  a  horse  for  $200,  a  dealer  loses  12J^%.  What 
would  he  have  gained  or  lost  per  cent  by  selling  at  $250? 

If  100  be  taken  to  represent  the  selling  price,  100  —  12J  =»  87J 
will  represent  the  cost, 

inn 

Therefore  the  cost  was  ^^  of  $200  =  $228f 
87j 

$250  -  $228^  =  $21f.  gain  by  selling  at  $250, 

Since  the  gain  on  $228;^  would  have  been  $21^,  the  gain  on  100 

would  have  been  ^  of  21f  =  9f.    That  is,  9f  %  gain.  Am. 


teachers'  edition.  293 

24.  A  spirit  merchant  buys  75  gals.,  at  ?3.25  a  gallon,  and,  after 
drawing  off  10  gals.,  sells  the  remainder  so  as  to  gain  5%  on  the 
whole.     What  is  the  selling  price  per  gallon  ? 

75  x|3.25  =  $243.75,  cost. 

If  100  be  taken  to  represent  the  cost,  100  +  5  =  105  will  repre- 
sent the  selling  price. 
Therefore,  the  selling  price  is  {^^  of  $243.75  =  $255.93f. 
75  gals.  —  10  gals.  =  65  gals.,  number  sold. 

^^^^•^^t  =  $3.93f,  selling  price  per  gallon.  Ans. 

25.  A  person  owns  two  estates  worth  respectively  |9845  and 
112,155.  If  the  first  rise  in  value  32%,  and  the  second  fall  13%, 
determine  the  rise  or  fall  per  cent  in  the  value  of  his  whole  property. 

If  100  be  taken  to  represent  the  value  of  the  first  estate  at  first, 
100  +  32  =  132  will  represent  its  value  after  rising  32%. 

Therefore,  the  value  after  rising  is  \^  of  $9845  -  $12,995.40. 

If  100  be  taken  to  represent  the  value  of  the  second  estate  at 
first,  100  —  13  =  87  will  represent  its  value  after  falling  13%. 

Therefore,  the  value  after  falling  is  y^^  of  $12,155  =  $10,574.85. 

$9845  +  $12,155  =  $22,000,  value  of  the  estates  at  first. 

$12,995.40  +  $  10,574.85  =  $  23,570.25,  value  afterwards. 

$ 23,570.25  -  $  22,000  =  $  1570.25,  actual  gain. 

Since  the  gain  on  $22,000  is  $1570.25,  the  gam  on  100  is  ^^^^-^ 
of  $1570.25  =  7^.     That  is,  7^^%  gain.  Ans. 

26.  A  tradesman  marks  an  article  $5,  but  takes  off  5%  for  cash. 
If  his  profit  is  14%,  what  was  the  cost  of  the  article? 

If  100  be  taken  to  represent  the  marked  price,  100  —  5  =  95 
will  represent  the  actual  price. 

Therefore,  the  actual  price  was  -\%^-  of  $5  =  $4.75. 

If  100  be  taken  to  represent  the  cost,  100  +  14  =  114  will  repre- 
sent the  price. 

Therefore,  the  cost  was  jff  of  $4.75  =  $4.16f .  Ans. 

27.  What  would  a  dishonest  dealer  gain  per  cent  by  using  a  false 
weight  of  15  oz.  instead  of  a  pound  ? 

16  oz.  —  15  oz.  =  1  oz.,  actual  gain. 

Since  the  gain  on  15  oz.  is  1  oz.,  the  gain  on  100  is  jV  o^  1^0  =  6f. 
That  is,  6f%.  Ans. 


294  ARITHMETIC. 


28.  A  dishonest  dealer  gains  12%  by  using  false  weights.  What 
is  the  real  weight  of  his  pound  ? 

If  100  be  taken   to  represent  the  false  weight  of  a  pound, 
100  +  12  =  112  will  represent  the   true  weight  of  a  pound. 
Therefore,  the  false  weight  is  {^^  of  16  oz.  =  14f  oz.  Ans. 

29.  A  tradesman,  in  selling  goods,  deducts  from  the  marked  price 
5%  for  cash.  What  is  the  marked  price  of  some  goods  for  which  he 
receives  $7.12^? 

If  100  be  taken  to  represent  the  marked  price,  100  —  5  =  95  will 

represent  the  actual  price. 
Therefore,  the  marked  price  is  -^  of  $  7.12^  =  $  7.50.  Am. 

30.  The  lead  ore  from  a  certain  mine  yields  60%  of  metal,  and  of 
the  metal  |  of  1%  is  silver.  How  much  silver  and  lead  will  be 
obtained  from  1200  tons  of  ore  ? 

If  100  be  taken  to  represent  the  ore,  60  will  represent  the  metal. 

Therefore,  the  amount  of  metal  is  j%\  of  1200  t.  =  720  t. 

If  100  be  taken  to  represent  the  metal,  |  will  represent  the  silver. 

Therefore,  the  amount  of  silver  is  -^  of  720  =  5.4  t.  Ans. 

100 
720  t.  -  5.4  t.  =  714.6  t.,  lead.  Ans. 

31.  If  ore  loses  41^%  of  its  weight  in  roasting,  and  43  J  %  of  the 
remainder  in  smelting,  how  much  ore  will  be  required  to  yield  1000 
tons  of  metal  ? 

If  100  be  taken  to  represent  the  ore,  100  —  41 J  =  58J  will  repre- 
sent the  ore  after  roasting. 
If  58 J  represent  the  ore  after  roasting,  58 J  —  43|%  of  58}  »  32f f 
will  represent  the  ore  after  smelting. 
3029 
Therefore,  -^  t.  will  be  the  amount  of  metal  in  1  ton  of  ore, 
100 

1000 
and  — — -  t.  =  3038.936  t.  will  be  the  amount  of  ore  required  to 
31|| 

100 

make  1000  tons  of  metal.  Ans. 


TEACHEKS'    EDITION.  295 

32.  How  many  per  cent  above  cost  must  a  man  mark  his  goods 
in  order  to  take  oflf  10%,  and  still  make  a  profit  of  17%? 

If  100  be  taken  to  represent  the  cost,  the  selling  price  will  be 

represented  by  117. 
As  the  selling  price  is  to  be  10%  below  the  marked  price,  the 

selling  price  (117)  will  be  j^^j-  of  the  marked  price. 
Therefore,  the  marked  price  will  be  \%^-  of  117  =  130. 
That  is,  the  goods  must  be  marked  30%  above  cost.  Ans. 

33.  How  many  per  cent  above  cost  must  a  man  mark  his  goods  in 
order  to  take  off  12^%,  and  still  make  a  profit  of  12^  %? 

If  100  be  taken  to  represent  the  cost,  the  selling  price  will  be 

represent-ed  by  1121. 
As  the  selling  price  is  to  be  12^  %  below  the  marked  price,  the 

selling  price  (112^)  will  be  — ^  of  the  marked  price. 

Therefore,  the  marked  price  will  be  - —  of  112J^  =  1284. 
That  is,  the  goods  must  be  marked  28|%  above  cost.  Ans. 

34.  How  many  per  cent  above  cost  must  a  man  mark  his  goods  in 
order  to  take  off  15%,  and  still  make  a  profit  of  15%? 

If  100  be  taken  to  represent  the  cost,  the  selling  price  will  be 

represented  by  115. 
As  the  selling  price  is  to  be  15%  below  the  marked  price,  the 

selling  price  (115)  will  be  -^^q  of  the  marked  price. 
Therefore,  the  marked  price  will  be  -y/  of  115  =  135/y. 
That  is,  the  goods  must  be  marked  35^^%  above  cost.  Ans. 

35.  How  many  per  cent  above  cost  must  a  man  mark  his  goods  in 
order  to  take  off  33.^  %,  and  still  make  a  profit  of  33^  %? 

If  100  be  taken  to  represent  the  cost,  the  selling  price  will  be 

represented  by  133|-. 
As  the  selling  price  is  to  be  33|-%  below  the  marked  price,  the 

selling  price  (133|-)  will  be  — ^  of  the  marked  price. 


—  of  133i 

66f 

That  is,  the  goods  must  be  marked  100%  above  cost.  Ans. 


Therefore,  the  marked  price  will  be  —  of  133|^  =  200 

66f 


296  ARITHMETIC. 


36.  If  5%  of  the  population  of  a  town  has  been  the  increase  in  the 
preceding  ten  years,  what  per  cent  of  the  population  ten  years  ago 
has  been  added  ? 

If  100  be  taken  to  represent  the  population,  the  population  10 

years  ago  will  be  represented  by  95. 
Since  the  gain  on  95  is  5,  the  gain  on  100  is  ^  of  5  =  5^^. 
That  is,  5^5^%.  Ans. 

37.  If,  in  a  population  of  27,000,000,  IS^/o  are  foreign-born,  how 
many  foreign-born  are  there  ?  What  is  the  ratio  of  the  foreign-born 
to  the  native  ? 

If  100  be  taken  to  represent  the  population,  13  will  represent 

the  foreign-born  population. 
Therefore,  the  foreign-born  population  is  y^^  of 

27,000,000  =  3,510,000  (1).        13% :  87%  =  13  :  87.  (2)  Ans. 

38.  A  man  bought  a  horse  for  $  70,  and  sold  him  for  $  80.  What 
per  cent  did  he  gain  ?  What  per  cent  of  the  money  received  for  the 
horse  was  gained  ? 

$80  -  $  70  =  $  10  actual  gain. 

Since  the  gain  on  $  70  is  $  10,  the  gain  on  100  is  ^  of  100  =  14f. 

That  is,  14f  %.  (1)  Ans. 

Since  the  gain  on  $80  is  $  10,  the  gain  on  100  is  |^  of  100  =  12J. 

That  is,  12^%.  (2)  Ans. 

39.  If,  by  selling  goods  for  12^,  per  cent  profit,  a  merchant  clears 
1 800,  what  was  the  cost  of  the  goods,  and  for  how  mucli  were  they 
sold? 

If  12i  represent  $800,  100  will  represent  ^  of  $800  =  $6400, 
cost.     $ 6400  -I-  $  800  =  $  7200,  selling  price.  Ans. 

40.  A  man  selling  eggs  at  40  cents  a  dozen  clears  33^  %  on  the 
cost;  what  was  the  cost?  Another,  selling  at  the  same  price,  clears 
33i  %  of  his  receipts  ;  what  did  his  eggs  cost  ? 

If  100  be  taken  to  represent  the  cost,  133}  will  represent  the 
selling  price. 


teachers'  edition.  297 

Therefore  the  cost  was of  40  cents  =  30  cents.  (1)  Ans. 

133i  ^  ^ 

If  100  be  taken  to  represent  the  selling  price,  66|  will  represent 

the  cost. 

Therefore,  the  cost  was  — ^  of  40  cents  =  26|  cents.  (2)  Ans. 
100  3  w 

41.  By  selling  a  carriage  for  $117,  a  carriage-maker  lost  10%  of 
the  cost.     What  ought  he  to  have  sold  it  for  to  make  10%? 

If  100  be  taken  to  represent  the  cost,  90  will  represent  the  sell- 
ing price. 

Therefore,  the  cost  was  -\%^-  of  $  117  =  1 130. 

If  100  be  taken  to  represent  the  cost,  1 10  will  represent  the  price 
at  which  he  ought  to  have  sold  it. 

Therefore,  the  selling  price  ought  to  have  been  {^^  of  $130  =  $143. 

42.  A  man  gained  in  January  3  %  in  weight,  and  in  February  lost 
3%.  What  per  cent  of  his  weight  on  the  first  day  of  January  is  his 
weight  on  the  first  day  of  March  ? 

If  100  be  taken  to  represent  his  weight  Jan.  1,  103  will  repre- 
sent it  Feb.  1. 

If  100  be  taken  to  represent  his  weight  Feb.  1,  97  will  repre- 
sent it  March  1. 

Therefore,  his  weight  March  1  was  y%V  ^^  ^^^  =  99tV^. 

99xV^  is  99tVit%  of  100.  Ans. 

43.  7  lbs.  of  a  certain  article  lose  3  oz.  in  weight  by  drying.  What 
per  cent  of  the  original  weight  is  water  ? 

7  lbs.  =  112  oz. 

If  100  be  taken  to  represent  the  whole  weight,  the  number  re- 
quired to  represent  3  oz.  is  y-f-Tj  of  100  =  2^f . 
That  is,  2|f%.  Ans. 

44.  7  lbs.  of  a  dry  article  have  lost  3  oz.  by  drying.  What  per 
cent  of  the  original  weight  was  water  ? 

7  lbs.  ==  112  oz.     112  oz.  -h  3  oz.  =  115  oz.,  whole  weight. 
If  100  be  taken  to  represent  the  whole  weight,  the  number  re- 
quired to  represent  3  oz.  is  jf  3^  of  100  ==  2^f . 
That  is,  2^1%.  Ans. 


298  ARITHMETIC. 


45.  A  dry  article  was  exposed  to  damp  air,  and  absorbed  3  oz.  of 
water ;  it  then  weighed  7  lbs.  What  per  cent  of  its  present  weight 
is  water  ? 

7  lbs.  =  112  oz. 

If  100  be  taken  to  represent  the  whole  weight,  the  number  re- 
quired to  represent  3  oz.  is  yf^^  of  100  =  2^|. 
That  is,  2^1%.  Am. 

46.  If  rosin  is  melted  with  20%  of  its  weight  of  tallow,  what  per 
cent  of  tallow  does  the  mixture  contain  ? 

If  100  be  taken  to  represent  the  rosin,  120  will  represent  the 

mixture. 
Therefore,  the  tallow  will  be  represented  by  -j^*^  of  100  =  16|. 
That  is,  16f%.  Ans. 

47.  If  20%  of  a  mixture  of  tallow  and  rosin  is  tallow,  what  per 
cent  of  the  weight  of  the  rosin  is  the  weight  of  the  tallow  ? 

If  100  be  taken  to  represent  the  mixture,  100  —  20  =  80  will 

represent  the  rosin. 
If  80  is  100%,  20  is  |g  of  100%  =  25%.  Ans. 

48.  How  many  pounds  of  tallow  must  be  mixed  with  8 J  lbs.  of 
rosin  in  order  that  the  mixture  may  contain  15%  of  tallow  ? 

If  100  be  taken  to  represent  the  mixture,  100  — 15  =  85  will 

represent  the  rosin. 
If  85  represent  8J  lbs.,  15  will  represent  |f  of  8J  lbs.  =  1 J  lb. 

49.  Nitrogen  gas,  under  standard  pressure  and  temperature,  is  \ 
of  1  %  of  the  weight  of  an  equal  volume  of  water.  What  is  its  spe- 
cific gravity? 

If  100  be  taken  to  represent  the  water,  |  will  represent  nitrogen. 

Therefore,  the  specific  gravity  of  nitrogen  is  -^  =  0.00125.  Ans. 

50.  Oxygen  gaa  is  }  of  1  %  of  the  weight  of  an  equal  volume  of 
water  ;  what  is  ita  specific  gravity  '!  How  many  gallons  of  oxygen 
will  it  take  to  weigh  as  much  as  a  pint  of  water?  How  many  of 
nitrogen  ? 


teachers'  edition.  299 


If  100  be  taken  to  represent  water,  |  will  represent  oxygen. 
Therefore,  tbe  specific  gravity  of  oxygen  is  -^— =  yi^.  (1)  Ans. 

As  1  gal.  of  oxygen  weighs  ^^^  of  a  gallon  of  water,  it  will  take 
as  many  gallons  of  oxygen  to  weigh  as  much  as  a  pint  of 

1 
water  as  -^—  =  87^  gals.  (2)  Ans. 

As  1  gal.  of  nitrogen  weighs  0.00125  of  a  gallon  of  water,  it  will 
take  as  many  gallons  of  nitrogen  to  weigh  as  much  as  a  pint 

of  water  as  — ^ =  100  gals.  (3)  Ans. 

0.00125  ^        ^  ^ 

51.  If  common  air  consist  of  4  volumes  of  oxygen  to  13  of  nitro- 
gen, what  is  its  specific  gravity  ? 

4  volumes  of  oxygen  ^^  4  X  j^^  =  xy3- 

13  volumes  of  nitrogen  =  13  X  -g-^^^  =  -gW. 

ih  +  jwu  =  3^w.  fo^  17  volumes. 

Therefore,  the  specific  gravity  is  -^^^^  -^  17  =  0.00129.  Ans. 

52.  How  many  gallons  of  air  weigh  as  much  as  a  pint  of  water  ? 
As  1  gallon  of  air  weighs  0.00129  of  a  gallon  of  water,  it  will 

take  as  many  gallons  of  air  to  weigh  as  much  as  a  pint  of 
i 

water  as s —  =  964-M  gals.  Ans. 

0.00129         ^'^  ^ 

53.  If  by  heating  iron  185°  F.  it  expands  |-  of  1  %,  what  will  be 
the  expansion  of  iron  in  passing  from  —  20°  F.  to  +  120°  F.  ? 

The  diiference  between  -  20°  and  +  120°  is  140°. 
If  I  of  1  %  is  the  expansion  for  185°,  the  expansion  for  140°  is 
iff  of  i  of  1%  =  ,75  of  1%.  Ans. 

54.  A  tubular  iron  bridge  is  450  ft.  long,  and  one  end  is  fast  to  a 
pier.  How  much  play  must  be  allowed  at  the  other  end,  if  the  iron 
expands  at  the  above  rate,  and  if  the  climate  varies  from  —  30°  F.  in 
winter  to  +  130°  F.  in  a  July  sun  ? 

The  difference  between  -  30°  and  +  130°  is  160°. 

If  I- of  1  %  is  the  expansion  for  185°,  the  expansion  for  160°  is 

llfofioflo/o  =  Aofl%. 
j\  of  1  %  of  450  ft.  is  if  ft.  =  5f|  in.  Ans. 


3CX)  ARITHMETIC. 


55,    How  much  longer  is  100  miles  of  iron  rail  at  118°  F.  than  at 
20°  below  zero? 

The  difference  between  -  20°  and  +  118°  is  138°. 

If  I  of  1  %  is  the  expansion  for  185°,  the  expansion  for  138°  is 

iMofioflo/o  =  /^^oflo/o. 
^  of  1  %  of  100  mi.  is  ^%  mi.  =  492^f  ft.  Am. 


Exercise  LXVIII. 

1.  Find  the  brokerage,  at  \  of  1%,  to  be  paid  on  1 10,450. 

$  10,450  X  0.00 J  =  1 13.06.  Ans. 

2.  Find  the  commission  on  $  2595,  at  2\  %. 

$2595  X  0.025  =  $64.88.  Ans. 

3.  An  agent  sells  200  bbls.  of  flour,  at  $6.25;  600  gals,  molasses, 
at  65  cents;  and  charges  a  commission  of  1|%.  What  are  the  net 
proceeds  ? 

200  x$  6.25  =  $1250.00 
600  X    0.65=      390.00 


$1640.00 
1640  X  0.01 1=        28.70 

$1611.30.  Ans. 

4.  A  commission  merchant  received  $1640  with  which  to  buy 
corn,  after  deducting  a  connnission  of  2^%.  What  is  the  amount  of 
his  commission,  and  how  many  bushels  of  corn  at  62 J  cents  a  bushel 
can  he  buy  ? 

If  100  be  taken  to  represent  the  amount  to  be  paid  for  the  goods, 
102^  will  represent  the  $1640.  Therefore,  the  amount  ex- 
pended for  goods  will  be  -^  of  $  1640  =  $  1600. 

And  2^%  of  $1600  =  $40,  commissions.  (1)  Ans. 

J^  bu.  -  2560  bu.  can  be  bought  for  $1600.  (2)  Ans. 
0.62^ 


teachers'  edition.  301 

5.  A  commission  merchant  sells  a  consignment  of  cotton  for  $5216. 
He  pays  $51  for  freight  and  storage,  and  charges  a  commission 
of  2^  %.     "What  are  the  net  proceeds  ? 

Consignment      =  $5216.00 

$5216x0.02^    =$117.36  =  commission. 

Storage  =      51.00 

Total  expenses  =  168.36 

$5047.64  =  net  proceeds.  Ans. 

6.  A  consignment  of  butter  was  sold  for  $  1570,  of  which  $  1546.45 
were  the  net  proceeds.     What  was  the  rate  per  cent  of  commission  ? 

$  1570  -  $  1546.45  =  $  23.55,  commission. 

If  100  be   taken   to   represent  the   consignment,  the   number 

23  55 
required  to  represent  $  23.55  will  be      ''  '    of  100  =  1|. 

1570 

That  is,  11%.  Ans. 
* 

7.  What  are  the  net  proceeds  from  the  sale  of  2250  bbls.  of  flour, 
at  $6.25  a  barrel,  if  the  charges  for  freight  and  storage  be  50  cents  a 
barrel,  commission  for  selling  2%,  for  guaranteeing  payment  1^%'^ 

2250  X  $6.25  =  $14062.50 

$14062.50  X  0.035  =  $492.19 

2250  X  $0.50  -  1125.00 

Total  expenses        =  1617.19 

$12445.31  =  net  proceeds.  Ans. 

8.  A  commission  merchant  sells  350  crates  of  peaches,  at  $2.60. 
If  the  commission  be  4|%,  find  the  net  proceeds. 

350  X  $2.60  =  $910.00 
$910x0.045=      40.95 

$869.05.  Ans. 

9.  A  man  sells  420  acres  of  land,  at  $40  an  acre,  and  charges  1^% 
commission.     What  is  his  commission? 

420  x$  40  =  $16,800 
$16,800x0.011  =  $210.  Ans. 


302  ARITHMETIC. 

10.  An  agent,  charging  4i%  commission,  receives  for  his  services 
$313.     Find  the  amount  of  his  sales. 

If  4J  represent  $313,  100  will  represent  —  of  $313  =  $6955.56. 

11.  A  merchant  buys,  through  an  agent,  730  yds.  of  carpeting,  at 
$1.2o  a  yard,  and  pays  the  agent  |  of  1%  commission  ;  the  freight 
amounted  to  $7.37.  At  what  price  per  yard  must  th§  carpeting  be 
sold  to  realize  a  profit  of  20%  ? 

730  x$  1.25  =  $91 2.50. 

0.0075  X  $912.50  =  $6.84,  commission. 

$912.50  +  $6.84  +  $7.37  =  $926.71,  total  expenses. 

If  100  represent  the  cost,  120  will  represent  selling  price. 

Therefore,  the  selling  price  must  be  \^  of  $926.71  =  $1112.05. 

$  11 1 2.05  -5-  730  =  $  1 .523.  Ans. 

/ 

12.  An  agent  sells  a  consignment  of  goods  for  $2100.     He  pays 

$33.50  for  freight,  and,  reserving  hia  commission,  remits  $2024.77. 
Find  the  rate  of  his  commission. 

$2024.77  +  $33.50  =  $2058.27,  consignment  less  commission. 
$2100  -  $2058.27  =  $41.73,  commission. 

If  100  be  taken  to  represent  the  consignment,  the  number  re- 
quired to  represent  $41.73  will  be  ^^  of  100  =  If^. 
That  is,  the  commission  was  lf§^%.  Ans. 

13.  A  commission  merchant  has  consigned  to  him  5000  lbs.  of 
cotton,  which  he  sells  at  14  cents  a  pound,  and  charges  2%  commis- 
sion. With  the  net  proceeds  he  buys  cotton  cloth,  at  10  cents  a  yard, 
charging  1^%  commission  for  buying.  How  many  yards  of  cloth 
does  he  buy  ? 

5000  X  $0.14  =  $700 
$700  X    0.02=      14^ 

$686  =  net  proceeds. 

If  100  be  taken  to  represent  the  amount  to  be  paid  for  the  goods, 
101.5  will  represent  the  $686. 


teachers'  edition.  303 


Therefore,  the  amount  expended  for  goods   will  be    of 

^  ^  101.5 

$686  =  $675.86. 

^^^  =  6758.6  yds.  can  be  bought  for  $675.86,  at  $0.10.  Ans. 

14.  A  commission  merchant  has  consigned  to  liim  500  bbls.  of  flour, 
which  he  sells  at  $5.50  a  barrel,  and  charges  2i%  commission  ;  the 
expenses  for  freight,  etc.,  amounted  to  $250.  Witli  the  net  proceeds 
he  buys  sugar,  at  6^  cents  a  pound,  charging  2^  %  commission  for 
buying.  How  much  sugar  does  he  buy,  and  what  is  the  amount  of 
his  commissions  ? 

500  x$  5.50      =  $2750.00 

$2750x0.025  =$68.75  =  commission. 

Freight,  etc.      =  250.00 
Total  expenses  =  318.75 

$2431.25  net  proceeds. 
If  100  be  taken  to  represent  the  amount  to  be  paid  for  the  sugar, 
2J  will  represent  the  commission,  and  102}  will  represent  the 
$2431.25.     Therefore,  the  amount  expended  for  sugar  will  be 

-1^  of  $  2431.25  =  $  2371.95. 
102^      *  * 

And  2^%  of  $2371.95  =  $59.30,  commission. 

Therefore,  MZL^^  lbs.  =  37,951.2  lbs.  will  be  bought.  (1)  Ans. 

$68.75 +  $59.30  =  $128.05,  whole  commission.  (2)  Ans. 

15.  A  collector's  commission  for  collecting  taxes,  at  1^%,  is 
$  206.55.     What  was  the  sum  collected  ? 

If  1 J  represent  $206.55, 100  will  represent  —  of  $206.55=$13,770. 

16.  An  agent  received  $2961  with  which  to  purchase  goods  after 
deducting  his  commission  at  5%.     How  much  was  his  commission? 

If  100  be  taken  to  represent  the  amount  to  be  paid  for  the  goods, 
5  will  represent  the  commission,  and  105  will  represent  the 
$2961.  Therefore,  the  amount  expended  for  the  goods  will  be 
l^f  of  $2961  =$2820. 

And  5%  of  $  2820  =  $  141,  commission.  Ans. 


304  ARITHMETIC. 


17.  An  agent  buys  3100  bbls.  of  flour,  at  $4.50  a  barrel,  and 
charges  1J%  commission.  What  is  the  amount  of  the  bill,  including 
the  commission  ? 

3100  X  $4.50  =  $13,950.00 
$13,950x0.015=         209.25 

$14,159.25.  Ans. 


18.  A  broker  receives  $6150  to  invest  in  cotton,  at  10}  cents  a 
pound.  His  commission  is  2J%.  How  many  pounds  of  cotton  can 
he  buy  ? 

Tf  100  be  taken  to  represent  the  amount  to  be  paid  for  the  cot- 
ton, 102J  will  represent  the  $6150.     Therefore,  the  amount 

expended  for  cotton  will  be  —  of  $6150  =  $6000. 

^^  lbs.  =  58,53614  lbs.  of  cotton  will  be  bought.  Ans. 
0.10}  "  ^ 


19.  An  agent  sells  1100  bbls.  of  flour,  at  $4.50  a  barrel,  and 
charges  2J%  commission.  He  invests  the  proceeds  in  steel,  at  IJ 
cents  a  pound,  ciiarging  1^%  commission.  What  is  his  entire  com- 
mission, and  how  many  tons  of  steel  (2240  lbs.  to  a  ton)  does  he  buy  ? 

1100  X  $4.50  =  $4950.00 
$  4950  X  0.025  =       1 23. 75  =  commission. 

$4826.25 

If  100  be  taken  to  represent  the  amount  to  be  paid  for  the  steel, 
IJ  will  represent  the  commission,  and  101.}  will  represent  tiie 
$4826.25.     Therefore,  the  amount  expended  for  steel  will  be 

^  of  $4826.25  =  $4754.93. 

And  1J%  of  $4754.93  =  $71.32,  commission. 

$123.75 +  $71.32  =  $195.07,  total  commission.  (1)  Ans. 

Therefore,  ^l^^f^  lbs.  =  316,995|  lbs.  =  141.51562  t.  of  steel 
will  be  bought.  (2)  Ans. 


teachers'  edition.  305 


Exercise  LXIX. 

1.  Find  the  premium  of  fire  insurance  for  $  2650,  at  |  of  1  %. 

1 2650  X  0.005  =  1 13.25.  Ans. 

2.  Find  the  premium  to  be  paid  for  insuring  a  person's  life  for 
$2500,  at  an  age  for  which  the  rate  is  2^%. 

$2500x0.02^  =  156.25.  Ans. 

3.  At  2|%,  what  premium  of  insurance  will  be  paid  on  a  vessel 
worth  $36,400? 

$  36,400  X  0.02f  =  $  1001.00.  Ans. 

4.  A  vessel  is  worth  $12,052.  Determine  t]ie  sum  to  be  insured, 
and  the  premium  to  be  paid  at  If  %,  so  that  in  the  event  of  loss  the 
owner  may  receive  both  the  value  of  the  vessel  and  the  premium. 

If  100  be  taken  to  represent  the  sum  to  be  insured.  If  will 
represent  the  premium,  and  100  —  If  ==  98^  will  represent  the 
value  of  the  vessel. 

Hence,  the  sum  to  be  insured  will  be 
$  12,052  ^  0.98^  =  $  12,266.67. 

And  If  %  of  $12,266.67  =  $214.67,  premium.  Ans. 

5.  The  premium  for  insurance  at  1|%  is  $150.  What  is  the 
amount  insured? 

If  11  represent  $  150,  100  will  represent  ^  of  $150  =  $12,000. 

6.  If  a  premium  of  insurance  at  2f  %  amount  to  $  28.60,  what  is 
the  sum  insured? 

If  2f  represent  $  28.60, 100  will  represent  ^  of  $28.60  =  $  1040. 

7.  A  vessel  is  so  insured  that  if  lost  the  owner  may  receive  both 
the  value  of  the  vessel  and  the  premium.  The  value  of  the  vessel 
is  $96,084,  and  the  rate  of  insurance  1|%.     Find  the  premium. 


306  ARITHMETIC. 


If  100  be  taken  to  represent  the  sum  to  be  insured,  IJ  will 
represent  the  promiurn,  and  100  —  1|  =  98J^  will  represent  the 
value  of  the  vessel. 

Hence,  the  sum  to  be  insured  will  be  $96,084  -i-  0.98  J  =  $97,920. 

And  1|  %  of  $97,920  =  $  1836,  premium.  Ans. 

8.  A  building  worth  $8000  is  insured  at  |  of  its  value,  at  |^  of  1% 
per  annum.     What  is  the  annual  premium? 

f  of  $8000  =  $5000. 
$5000x0.00^  =  $6.25.  Ans. 

9.  Four  companies  join  in  insuring  a  ship  and  cargo  for  $60,000. 
One  company  takes  ^,  at  f  of  1  % ;  a  second  takes  $  10.000,  at  f  of  1  % ; 
a  third,  $15,000,  at  |  of  1%;  a  fourth,  the  remainder,  at  ^  of  1%. 
How  much  is  paid  for  insurance  ? 

(1)  (2)                 (3)                        (4) 

^  of  $60,000  =  $20,000.  $10000  $15000  The  remainder  is 

$20000  O.OOf            O.OOf               $15000 

•      O.OOf  .T--.  ITTZ:                 O.OOi 


$75.00  $93.75 


$120.00  $75.00 

$;20  +  $75  +  $93.75  +  $75  =  $363.75,  total  premium.  Am. 

10.   If  the  ship  in  the  last  problem  receive  damage  to  the  amount 
of  $4500,  what  ought  each  company  to  pay  ? 

^  of  $4500  =  $1500  (1). 
mU==l    i  of  $4500  =  $750  (2). 
Mm  =  i-    i  of  $4500  =  $  1125  (3).  (4). 


11.  A  man  insures  his  life  for  $10,000,  paying  $350  a  year  in 
advance.  He  dies  the  day  before  the  fifth  premium  was  due.  The 
company  pay  his  widow  $10,000.  How  much  have  they  lost  by 
him,  if  the  interest  gained  on  the  premiums  paid  amount  to  $175' 

4  X  $350  +  $  175  =  $  1575,  total  premiums  with  interest. 
$  10,000  -  $  1575  =  $  8425,  company's  loss.  Ans. 


teachers'  edition.  307 

12.  A  merchant  shipped  a  cargo  to  London ;  and  to  cover  both 
the  cargo  and  the  premium,  he  took  out  a  policy  of  $  100,800,  at  3|  %. 
What  was  the  value  of  the  cargo  ? 

If  100  be  taken  to  represent  the  sum  insured,  100  —  3|  =  96| 

will  represent  the  value  of  the  cargo. 
Hence,  the  value  of  the  cargo  is  0.96i  of  1 100,800  =  $97,272. 

13.  Three  companies  insure,  at  f  of  its  value,  a  building  worth 
1 16,000.  The  first  company  takes  i  the  risk,  at  |  of  1  % ;  the  second, 
f  of  it,  at  I  of  1  % ;  and  the  third,  the  remainder,  at  f  of  1  %.  Find 
the  total  premium. 

f  of  $16,000 -$12,000. 
^  of  $  12,000  =  $4000.  f  of  $  12,000  -  $4800. 

$4000  $4800 

0.00^  0.001 


$30.00  $42.00 

$  12,000  -  $  4000  -  $  4800  =  $  3200. 
$3200 
O.OOf 

$24.00 

$30  +  $42  +  $24  -  $96,  total  premium.  Ans. 

14.  S.  Williams  pays  $  18.40  premium  for  insuring  his  house  for 
f  of  its  value,  at  \\%.    What  is  the  value  of  his  house? 

If  11  represent  $  18.40, 100  will  represent  ^  of  $  18.40  =  $  1226,2. 

If  $  1226|  is  I,  the  whole  is  f  of  $  1226|  =  $  1840.  Ans. 

Exercise  LXX. 

1.  If  James  Brown  be  assessed  $2500  on  his  house,  and  $5200  on 
personal  property,  and  pays  for  2  polls  at  $  1.50  each,  how  much  will 
his  tax  be,  the  rate  being  $  12.18  on  $  1000? 

$2500 +  $5200  =  $7700. 
0.01218  of  $7700  =  $93.79. 
$93.79 +  $3.00  =  $96.79.  Ans. 


308  ARITHMETIC. 


2.  If  the  rate  of  tax  be  $  12.25  on  $  1000,  and  the  tax  be  $11,788.50, 
what  is  the  valuation  ? 

112.25  on  $1000  =  1.225%. 

If  1.225  represent  $  11,788.50,  100  will  represent  ^^   of 

$11,788.50  =  $962,326.53.  Ans. 

3.  If  the  assessed  valuation  of  a  town  be  $1,777,000,  and  the 
property-tax  be  $6870,  what  is  the  rate  on  $1000? 

$6870  ^  $  1,777,000  =  0.003866. 

That  is,  the  rate  is  0.3866  of  1%,  or  $3,866  on  $1000.  Ans. 

4.  What  sura  must  be  assessed,  in  order  that  $15,000  shall  remain 
after  paying  a  commission  of  2%  for  collecting  the  taxes? 

If  100  be  taken  to  represent  the  sum  to  be  assessed,  98  will  rep- 
resent the  $  15,000. 
Therefore,  the  sum  to  be  assessed  is  -V^-  of  $  15,000  =  $  15,306.12. 

5.  A  tax  of  $  1857.60  is  levied  upon  a  school  district  for  building 
a  school-house.  The  assessed  valuation  of  the  district  is  $1,935,000. 
What  is  the  tax  on  property  valued  at  $6250? 

$  1857.60  ^  $  1,935,000  =  0.00096. 

Therefore,  tax  on  the  property  is  0.00096  of  $6250  =  $6.00.  Ans. 

6.  In  a  certain  town  there  are  1350  polls.  The  assessed  value  of 
the  real  estate  is  $713,250;  of  the  personal  property  is  $738,954; 
the  poll-tax  is  $2.  The  tax  on  property  is  l|%.  But  only  96%  of 
the  tax  can  be  collected,  and  the  collector  is  paid  2|%  of  the  amount 
collected.     How  much  does  the  town  receive  from  the  taxes  ? 

The  amount  of  poll-taxes  =  1350  X  $2  =  $2700. 
$713,250  -f  $738,954  =  $1,452,204,  assessed  value. 
0.01^  of  $1,452,204  =  $16,337.31,  amount  to  be  raised  on  prop- 
erty. 
$16,337.31  4  $2700  =  $19,037.31,  amount  to  be  raised. 
0.96  of  $  19,037.31  =  $  18,275.82,  amount  collected. 
0.02i  of  $  18,275.82  =  $456.90,  collector's  pay. 
$18,275.82  -  $456.90  -  $  17,818.92,  amount  town  receives.  Ana. 


teachers'  edition.  309 


7.  What  is  the  duty,  at  20%  ad  valorem  (that  is,  20%  of  the  cost), 
on  320  boxes  of  raisins,  each  containing  40  lbs.,  and  costing  8  cents 
a  pound  ? 

320  X  40  x$  0.08  =  $1024.00. 
0.20  of  $  1024  =  $  204.80.  Ans. 

8.  What  is  the  duty,  at  6  cents  a  gallon,  on  420  hhds.  of  molasses, 
63  gals,  in  a  hogshead  ? 

420  X  63  X  I  O.OG  =  %  1587.60.  Ans. 

9.  At  40%,  what  is  the  duty  on  300  tons  of  iron  (2240  lbs.  to  a 
ton)  invoiced  at  1\  cents  a  pound  ? 

300  X  2240  X  $  0.01 1  =  1 10,080. 

0.40  of  1 10,080  =  $  4032.  Ans. 

10.  Paid  $1360.80  duty  on  300  hhds.  of  molasses,  each  containing 
63  gals.,  at  25  cents  a  gallon.     Wliat  was  the  rate  per  cent  of  duty? 

300x63x$0.2o  =  $4725. 
If  100  be  taken  to  represent  $4725,  the  number  required  to 

represent  $  1360.80  is  ^-^7^  ^^  ^^^  =  28f . 

That  is,  28|%.  .4ns. 

11.  A  sugar  refiner  imports  50  hhds.  of  sugar  weighing  480  lbs. 
each,  and  120  hhds.  of  molasses  containing  63  gals.  each.  What  is 
the  amount  of  the  duties,  if  the  sugar  pay  3  cents  a  pound,  and  the 
molasses  8  cents  a  gallon,  an  allowance  being  made  on  the  sugar  of 
10%,  and  2%  on  the  molasses? 

50  X  480  x$  0.03  =  $720. 
0.00  of  $720  =  $648. 
120  X  63  x$  0.08 -$604.80 
0.98  of  $604.80  =  $592.70. 

+  $592.70  =  $1240.70.  Ans. 


12.  An  importer  paid  $825  duty  on  an  invoice  of  silks,  the  duty 
being  24%.  But  damages  of  15%  were  allowed  at  the  custom-house. 
What  was  the  entire  cost  of  the  goods  ? 


310 


ARITHMETIC. 


If  24  represent  |825,  100  will  represent  -^^  of  $825  =  $3437.50. 
If  100  be  taken  to  represent  the  cost,  100  —  15  =  85  will  repre- 
sent the  invoice  price. 
J^/  of  $3437.50 -$4044.12,  cost. 
$4044.12 +  $825  =  $4869.12.  Ans. 

13.  Paid  $  325  duty  on  goods  which  had  been  damaged ;  allow- 
ance for  damage  is  24%,  and  the  duty  was  24%.  What  was  the 
invoice  price  of  the  goods  ? 

If  24  represent  $325, 100  will  represent  ^^  of  $325  =  $1354.17. 
If  100  be  taken  to  represent  the  cost,  100  —  24  =  76  will  repre- 
sent the  invoice  price. 
Therefore,  the  invoice  price  was  -W"  of  $1354.17  -  $1781.80. 

Exercise  LXXI. 


1.  Find  the  interest  of  $080.40 
for  2  yrs.  4  mos.  6  dyp.,  at  6%. 

2  yrs.  =  0.12 
4  mos.  =  0.02 
6  dys.  =  0.001 

0.141 

$680.40 
X  0.141 

$95.94.  Ans. 

2.  Find  the  interest  of  $25,625 
for  30  dys.,  at  6%. 

30  dys.  =  0.005. 
$25,625 
X  0.005 


$0.13.  Ans. 

3.  Find  the  interest  of  $85.85 
for  1  yr.  7  mos.  21  dys.,  at  6% 


1  yr.  =  0.06 
7  mos.  =  0.035 
21  dys.  =  0.0035 

0.0985 

$85.85 
X  0.0985 

$8.46.  Ans. 

4.   Find  the  interest  of  $1100 
for  3  yrs.  4  mos.,  at  5%. 

3  yrs.  =0.18 

4  mos.  =  0.02 

020 

5 

6)1.00 
O.lf 

$1100 

xo.if 

$183.33.  Am. 


TEACHERS     EDITION. 


311 


5.    Find  the  interest  of  $1275 
for  3  yrs.  2  mos.  15  dys.,  at  8%. 

3  yrs.  =  0.18 
2  mos.  =  0.01 
15  dys.  =  0.0025 
0.1925 
4 


3)0.7700 


11275 

X  0.25f 

$327.25.  Ans. 

6.   Find  the  interest  of  $475.16 
for  27  dys.,  at  4i%. 

27  dys.  =  0.0045 
X3 


4)0.0135 


$475.16 
X  0.0033f 

$1.60.  Ans. 


7.   Find  the  interest  of  $1290.50 
for  60  dys.,  at  6%. 

60  dys.  =  0.01. 
$1290.50 
XO.OI 


$12.91.  Ans. 

8.   Find  the  interest  of  $125 
for  1  yr.  2  mos.  2  dys.,  at  9%. 


1  yr.  =  0.06 
2  mos.  =  0.01 
2  dys.  =  0.0001- 
0.070i 
X3 


2)0.211 
0.1055 
X$125 
$13.19.  Ans. 

9,  Find  the  interest  of  $250.80 
for  10  mos.  10  dys.,  at  31%. 

10  mos.  =  0.05 
10  dys.  =  0.00 If 
0.05  If 

7_ 

12)  0.36  If 
0.030/^ 

$250.80 
X  0.030/^ 
$7.56.  Ans. 

10.  Find  the  interest  of 
$258.85  from  Mar.  6  to  June  24, 
at  5%. 

mos.  dys. 

6  24 

3  6 


3 

18 

3 

mos 

=  0.015 

18 

dys 

=  0.003 
6)0.018 

0.003 
X5 

0.015 

$258.85 

X 

0.015 

$3. 


312 


ARITHMETIC. 


11.  Find  the  interest  of  f  380 
for  2  yrs.  11  raos.  27  dys.,  at  4^%. 

2yr8.  =0.12 
11  nios.  =0.055 
27  dys.  =  0.0045 

0.1795 
X3 

4)0.5385 
0.1 346^ 
X|380 

$51.16.  Ans. 

12.  Find  the  interest  of 
$475.05  for  1  yr.  9  mos.  14  dys., 
at  7xV%. 

1  yr.  =  0.073 
9  mos.  =  0.05475 
14  dys.  =  0.002838 

0.130588 
X  $475.05 

$62.04.  Ans. 

13.  Find  the  interest  of 
$725.40  for  11  mos.  24  dys., 
at  5^0/0. 

11  mos.  =  0.0481  J 
24  dys.  =  0.0035 

0.0516^ 

$725.40 
X  0.0516^ 

$37.45.  Ans. 

14.  Find  the  interest  of 
$680.50  for  2  yrs.  6  days,  at  5%. 


$680.50 
XO.05 

$34.0250  int.  1  yr. 
$68.62.  Ans. 

15.  Find     the      interest      of 
$630.50  for  90  dys.,  at  6%. 

90  dys.  =  0.01^. 
$630.50 
X  O.OH 
$9.46.  Am. 

16.  Find     the     interest     of 
$547.60  from  Feb.  20  to  Dec.  5, 

at6|%. 


12 

2 


dys. 

5 

20 


9 


15 


9  mos.  =  0.045 
15  dys.  =  0.0025 

6)0.0475 
0.0079^ 
X6^ 

0.05145f 

$547.60 
X  0.05145f 

$28.18.  Ans. 

17.  Find  the  interest  of  $875 
from  May  5,  1880,  to  June  21, 
1881,  at5i%. 


TEACHERS     EDITION. 


313 


jrs. 
1881 

1880 


dys. 

21 
5 


1  1  16 

1  yr.  =  0.06 

1  mo.  =  0.005 

16  dys.  =  0.002f 

12)0.0671 
0-005|f 

0.062^V 

1875 
X  0.O623V 

154.27.  Ans. 


18.  Find  the  interest  of 
$758.50  from  Jan.  5  to  July  1, 
at4i%. 


dys. 
1 

5 


5  26 

5  mos.  =  0.025 
26  dys.  =  0.004^ 

4)0.029^ 
0.007^ 

0.022 

$758.50 
X  0.022 

$16.69.  Ans. 

19.  Find  the  interest  of 
$342.42  from  Feb.  5,  1879,  to 
Mar.  15,  1881,  at  7%. 


yrs. 
1881 

mos. 

3 

dys. 

15 

1879 

2 

5 

2 

1 

10 

2 
1 

yrs.  =  0.12 
mo.  =  0.005 

10 

dys.  =  O.OOlf 
6)0.12f 
0.02^ 
0.14| 

$342.42 
X  0.141 
$50.60.  Ans. 

20.   Find  the  interest 

of  $540 

from  Mar. 

5  to  Sept.  21, 

at  310/0- 

mos.                        dys. 

21 
5 


6  16 

6  mos.  =  0.03 
16  dys.  ==  0.0021 
0.032f 

7^ 

12)0.2281 
0.019xV 
$540 
x0.019t-V 
$10.29.  Ans. 

21.   Find      the      amount     of 
$431.50  for  2  yrs.  8  mos.,  at  ^% 
$431.50 

x  0.041 

$19.41 75  =  Int.  for  1  yr. 
X  2|  =  2  yrs.  8  mos. 
$51.78 
431.50 


$483.28.  Ans. 


314 


ARITHMETIC. 


22.  Find  the  amount  of 
$476.50  from  July  5,  1880,  to 
Feb.  9,  1881,  at  4%. 


yrs.                  mos.             dys. 

1881            2            9 
1880            7            5 

yrs. 

1880 
1878 

1 

lyr. 
10  mos. 
22  dys. 

mos.              dys. 

5  7 

6  15 

7            4 
7  mos.  =  0.035 
4  dys.  =  O.OOOf 

3)0.035| 

10          22 
=  0.06 
=  0.05 
=  0.003f 

O.Ollf 


0.023J 
1476.50 
X  0.023^ 

$11.33 
476.50 


$487.83.  Ans. 

23.  Find  the  amount  of 
$310.20  from  April  7  to  Aug.  31, 
at  3^0/,. 


dTS. 

31 

7 


4  24 

4  mos.  =  0.02 
24  dys.  =  0.004 


6)0.024 
0.004 

H 

0.013 

$319.20 

X  0.013 


$4.15 
319.20 


$323.35.  Ans. 


24.  Find  the  amount  of  $  6460 
from  June  15,  1878,  to  May  7, 
1880,  at  4^%. 


6)  0.1131 
0.0l8f| 
X4.25 

0.0805  ItV 
X$6460 

$520.12 
6460.00 


12.  Ans. 


25.   Find  the  amount  of  $150 
from  Aug.  5,  1879,  to  Mar.   17, 
1881,  at  7%. 

yrs.                 mos.                dys. 

1881            3            17 
1879            8              5 

1            7 

lyr.  =  0.06 
7  mos.  =  0.035 
12  dys.  =  0.002 

12 

6)0.097 
0.0161 

O.llSi 
X$150 

$16.98 
150.00 

$166.98.  Ans. 


TEACHERS     EDITION. 


315 


26.   Find     the 
$527.20  from  Jan. 
at  410/0. 


mos. 
11 


amount      of 
1  to  Nov.  20, 


dys. 

20 

1 


10 
10  mos. 
19  dys. 


19 
=  0.05 
=  0.003^ 

4)0.053^ 
0.0133s^ 

0.0391 
$527.20 
X  0.0391 

$21.02 
527.20 


$548.22.  Ans. 

27.  Find  the  amount  of  1 1250 
from  Nov.  15,  1880,  to  Mar.  1, 
1881,  at  5%. 


yrs. 

1881 
1880 


mo». 

3 
11 


dys. 
1 

15 


3  16 

3  mos.  =  0.015 
16  dys.  =  0.002f 


6)0.017f 
0.002i| 
0.014}f 
$1250 
X  0.01411 

$18.40 
1250.00 


$1268.40.  Ans. 


28.  Find  the  amount  of 
$624.36  from  Mar.  5  to  Dec.  20. 
at  IjV/o- 


mos. 

dys. 

12 

20 

3 

5 

9 

15 

=  9^ 

mos. 

$624.36 

X  0.073 

12)145.57828.  Int.  for  1  yr. 
$3.79819.  Int.  for  1  mo. 
X9^ 

$36.08 
624.36 


,44.  Ans. 


29.   Find     the 
$12,260  from  May 
at  3f  %. 


mos. 

10 


amount     of 
6  to  Oct  24, 

dys. 

24 
6 


5  18 

5  mos.  =  0.025 
18  dys.  =  0.003 

0.028 
X3| 


6)0.105 
0.0175 
$12260 
X  0.0175 

$214.55 
12260.00 

$12,474.55.  Ans. 


316  ARITHMETIC. 


mos. 
12 

10 

dy.. 
31 

20 

2 

11 

2  mos. 
11  dys. 

=  0.02 
=  O.OOSf 

30.  Find  the  amount  of  1 11,216  from  Oct.  20  to  Dec.  31,  at  1% 
a  month. 

$11216 
X  0.023f 

12(55.45 
11216.00 

$11,481.45.  Am. 
0.023f 

31.  P'ind  the  rate  per  cent  when  the  interest  on  $326  for  15  yre. 
is  $220.05. 

Interest  on  $326  for  15  yrs.  is  $202.05; 

on  $326  for  1  yr.  is  j\  of  $202.05  ; 

on  $  1  for  1  yr.  is  ^^^  of  yV  of  $202.05  =  $0.04^. 
Therefore,  the  rate  required  is  4|%.  Ans. 

32.  Find  the  rate  per  cent  when  the  interest  on  $372.50  for  18  yre. 
is  $301,725. 

Interest  on  $372.50  for  18  yrs.  is  $301,725  ; 
on  $372.50  for  1  yr.  is  l-^^-Z^^ . 

on  $1  for  1  yr.  is    ^^^^■'^^^    =  $0.04^ 
^  ^         18x372.50  ^ 

Therefore,  the  rate  required  is  4|%.  Am. 

33.  Find  the  rate  per  cent  when  $245  amount  to  $252.96^  for 
9  mos. 

The  interest  is  $  252.96^  -  $  245  =  $  7.96^. 

The  time  is  9  mos.  =  0.75  yr. 

Interest  on  $245  for  0.75  yr.  is  $  7.96 J ; 

on$245forlyr.  is^^^; 
Therefore,  the  rate  required  is  4  J  %.  Am. 


teachers'  edition.  317 

34.  Find  the  rate  per  cent  when  the  interest  on  $  235.25  is  1 70.575 

for  5  yrs. 

Interest  on  $235.25  for  5  yrs.  is  $  70.575  ; 

on  1235.25  for  1  yr.  is  lIMTi^ . 
5 

on  $1  for  1  yr.  is    ^'^^'^^^    =  $0.06. 
^  ^         5x235.25      ^ 

Therefore,  the  rate  required  is  6  %.  Ans. 

35.  Find  the  rate  per  cent  when  $363,125  amount  to  $371,598 
for  7  mos. 

The  interest  is  $  371.598  -  $  363.125  =  $  8.473. 
The  time  is  7  mos.  =  /^  yr- 
Interest  on  $  363. 1 25  for  y^  yr-  is  $  8-473  ; 
on  $363,125  for  1  yr.  is  ^MZ^  . 

on  $  1  for  1  yr.  is ^^ =  $0.04,  nearly. 

*  ^         T^^X  $363,125      ^  ^ 

Therefore,  the  rate  required  is  4%,  nearly.  Ans. 

36.  Find   the  rate  per  cent  when   the  interest  on  $249.43|  is 
$49.88|  lor  5  yrs.  4  mos. 

The  time  is  5  yrs.  4  mos.  =  5^  yrs. 
Interest  on  $249.43f  for  5i  yrs.  is  $49.88f  ; 

on  $249.43f  for  1  yr.  is  ^^^1^ 

Therefore,  the  rate  required  is  3|%.  Ans. 

37.  Find  the  rate  per  cent  when  $  350  amount  to  $406.70  for  3  yrs. 
7  mos.  6  dys. 

The  time  is  3  yrs.  7  mos.  6  dys.  =  3.6  yrs. 
The  interest  is  $406.70  -  $350  =  $56.30. 
Interest  on  $350  for  3.6  yrs.  is  $56.30; 
56.30 


on  $  350  for  1  yr.  is 


3.6 


Therefore,  the  rate  required  is  4^  %.  Ans. 


318  ARITHMETIC. 


38.  Find  the  rate  per  cent  when  the  interest  on  $6875  is  1 72.05 
for  90  dys. 

The  time  is  90  dys.  =  0.25  yr. 

Interest  on  $6875  for  0.25  yr.  is  $72.05; 

on$6875forlyr.  isi^; 

on$lforlyr.is^|^  =  $0.04^^. 
Therefore,  the  rate  required  is  41  %.  nearly.  Ans. 

39.  Find  the  rate  per  cent  when  the  interest  on  $642  is  $10.70 
for  5  mos. 

The  time  is  5  mos.  =  ^^  yr. 

Interest  on  $642  for  ^V  yr-  is  $10.70; 

on  $642  for  lyr.  is  5^^:^; 

on  $  1  for  1  yr.  is  -IIM^.  =  |o.04. 
^    _     AX642 

Therefore,  the  rate  required  is  4  %.  Ans. 


40.  Find  the  rate  per  cent  when  the  interest  on  $  8432  for  2  yrs. 
7  mos.  23  dys.  is  $  1339.28. 

The  time  is  2  yrs.  7  mos.  23  dys.  =  2f|§  yrs. 
Interest  on  $8432  for  2f|^  yrs.  is  $1339.28 ; 

on$8432forlyr.  isil^^^; 

on  $1  for  1  yr.  is  gf^^f^s^o  ^  ^  ^•^^'  "'*'^^- 
Therefore,  the  rate  required  is  6%,  nearly.  Ans. 

41.  Find  the  rate  per  cent  when  a  sum  of  money  is  doubled  in 
14  yrs. 

Interest  on  $1  for  14  yrs.  is  $1 ; 

on$lforlyr.  is^  =  $0.07f 

Therefore,  the  rate  required  is  7i  %   Ans. 


teachers'  edition.  .  319 

42.   Find  the  rate  per  cent  when  an  investment  for  5  yrs.  2  mos. 
produces  a  sum  equal  to  f  of  the  capital. 

The  time  is  5  yrs.  2  mos.  =  51  years. 
Interest  on  1 1  for  5^  yrs.  is  2_ ; 


II 
Therefore,  the  rate  required  is  7|f  %.  Ans. 


on$lforlyr.  is  J5  =  $0.07ff 


43.   Find  the  rate  per  cent  when  an  investment  for  3  yrs.  1  mo. 
15  dys.  produces  a  sum  equal  to  ^  of  the  capital. 

The  time  is  3  yrs.  1  mo.  15  dys.  =  3|  yrs. 

$1 
Interest  on  1 1  for  3|-  yrs.  is  ^  ; 
8 

on  $1  for  lyr.  is  ^  =  10.04. 
Therefore,  the  rate  required  is  4%.  Ans. 


44.  Find  the  time  in  which  the  interest  on  $450  will  amount  to 
$72,  at  4%. 

Interest  on  $450,  at  4%,  for  1  yr.  is  0.04  of  $450  =  $18. 

Therefore,  the  number  of  years  will  be  |—  =  4  yrs.  Ans. 

$18 

45.  Find  the  time  in  which  the  interest  on  $487.50  will  amount 
to  $39,  at  4%. 

Interest  on  $487.50,  at  4%,  for  1  yr.  is  0.04  of  $487.50 -$19.50. 
Therefore,  the  number  of  years  will  be    ';        =  2  yrs.  Ans. 

46.  Find  the  time  in  which  the  interest  on  $238.75  will  amount 
to  $64.46,  at  4^%. 

Interest  on  $238.75,  at  4|%,  for  1  yr.  is  0.045  of  $238.75  =  $10.74. 

Therefore,  the  number  of  years  will  be  ^    '     =  6  yrs.  Ans. 


320  ARITHMETIC. 


47.  Find  the  time  in  which  the  sum  of  $  793,875  will  amount  to 

$805.84,  at  5^%. 

The  interest  is  $  805.84  -  $  793.875  -  $  11.965. 
Interest  on  $793,875,  for  1  yr.  at  5^%  is  0.055  of 
$793,875  =  143.663. 

Therefore,  the  number  of  years  will  be  Z. — '■ —  =  0.274. 
^  $43,663 

0.274  yr.  =  3  mos.  9  dys.  Ans. 

48.  Find  the  time  in  which  a  sum  of  money  will  double   itself 
at  40/0. 

Interest  on  $1,  at  4%,  for  1  yr.  is  0.04  of  $1  =$0.04. 

$1 
Therefore,  the  number  of  years  will  be  -^ —  =  25.  Ans. 
^  $0.04 

49.  Find  the  time  in  which  the  sum  of  $10  will  amount  to  $17, 
at  6%. 

The  interest  is  $  1 7  -  $  10  =  $  7. 

Interest  on  $  10,  at  6%,  for  1  yr.  is  0.06  of  $10  =  $0.60. 

$7 
Therefore,  the  number  of  years  will  be  -^ —  =  11|. 
•^  $0.60 

llf  yrs.  =  11  yrs.  8  mos.  Ans. 

50.  Find  the  time  in  which  the  sum  of  $502.67  will  amount  to 

$578.07,  at  4|%. 

The  interest  is  $587.07  -$502.67  =  $75.40. 

Interest  on  $502.67,  at  4|%,  for  1  yr.  is  0.045  of  $502.67  =  $22.62. 

$75  40 
Therefore,  the  number  of  years  will  be  ^     '      =  3^. 

$  22.62 

3^  yrs.  =  3  yrs.  4  mos.  Ans. 

51.  Find  the  time  in  which  the  interest  on  $537.50  will  amount 
to  $80,625,  at  4%. 

Interest  on  $537.50,  at  4%,  for  1  yr.  is  0.04  of  $537.50  =  $21.50. 

S  80  625 
Therefore,  the  number  of  years  will  be  *-  '  -  -  =  3.75. 

^  Jl.oU 

3.75  yrs.  -  3  yrs.  9  mos.  Ana. 


teachers'  edition.  321 

52.  Find  the  time  in  which  the  interest  on  |6875  will  amount 
to  $75.05,  at  41% 

Interest  on  |6875,  at  4^%,  for  1  yr.  is  0.0425  of  $6875  =  1 292.19. 

%  75  05 
Therefore,  the  number  of  years  will  be  '*^   '  ' —  =  0.256. 
^  $292.19 

0.256  yrs.  =  3  raos.  2  dys.  Arts. 

53.  Find  the  time  in  which  the  interest  on  $8520  will  amount 
to  $1746.60,  at  6%. 

Interest  on  $8520,  at  6%,  for  1  yr.  is  0.06  of  $8520  =  $511.20. 

Therefore,  the  number  of  years  will  be ' —  =  SyV 

^  $511,20        ^^ 

3^^  yrs.  =  3  yrs.  5  mos.  Ans. 

54.  Find  the  principal  that  will  produce  $90  interest  in  3  yrs., 
at  4%. 

Interest  for  1  yr.  is  - —  =  $30. 

Interest  on  $1  for  1  yr,,  at  4%  =  $0.04. 

30 
Hence,  principal  required  =  — —  of  $  1  =  $  750.  Ans. 

55.  Find  the  principal  that  will  produce  $  63  interest  in  3  yrs., 
at  6i%. 

Interest  for  1  yr.  is  - —  =  $21. 

Interest  on  $1  for  1  yr.,  at  6^%  =  $0.0625. 

21 

Hence,  principal  required  = of  $1  =  $336.  Ans. 

^        ^         ^  0.0625      ^         ^ 

56.  Find  the  principal  that  will  produce  $  100  interest  in  8  yrs. 
6  mos.,  at  5%. 

8  yrs.  6  mos.  =  8.5  yrs. 

Interest  for  1  yr.  =  ^1^  =  $11.7647. 
•^  8.5 

Interest  on  $  1  for  1  yr.,  at  5%  =  $0.05. 

Hence,  principal  required  =  ILj!}^  of  $1  =  $235.29.  Ans. 


322  ARITHMETIC. 


57.  Find  the  principal  that  will  produce  $1746,60  interest  in 
3  yrs.  5  mos,,  at  6%. 

3  yrs.  5  mos.  =  3^^  yrs. 

Interest  for  1  yr.  =  llZi^  =  $511.20. 

Interest  on  |1  for  1  yr.,  at  6%,  is  $0.06. 

Hence,  principal  required  =  — ^^  of  $1  =$8520.  Am. 

58.  Find  the  principal  that  will  produce  $  12  interest  in  7  mos., 
at  5%. 

7  mos,  =  3^  yr. 

Interest  for  1  yr.  =  ^  =  $20.5714. 

Interest  on  $1  for  1  yr.,  at  5%  =  $0.05. 

20  5714 
Hence,  principal  required  =    ^'         of  $1  =  $411,43.  Ans. 

59.  Find  the  principal  that  will  produce  $50  interest  in  228  dys., 
at4|%. 

228  dys.  =  H  yr- 

Interest  for  1  yr.  =  ^  =  $78,9474, 

Interest  on  $1  for  1  yr.,  at  4^%  =  $0,045. 

78  *^474 
Hence,  principal  required  =  — ^^^ of  $  1  =  $  1754.39.  Ans. 

60.  Find   the   principal   that  will  produce  $1339.28  interest  in 
2  yrs.  7  mos.  24  dys,,  at  6%. 

2  yrs.  7  mos,  24  dys.  =  2.65  yrs. 

Interest  for  1  y r.  =  ^^^^^'^^  -  $ 505.3887. 
^  2,65 

Interest  on  $1  for  1  yr,,  at  6%  =  $0.06. 

Hence,  principal  required  =  ^^'^^^"^  of  $  1  =  $8423.14.  Ans. 


teachers'  edition.  323 

61.  Find   the  principal   that   will  produce  $1312.65  interest  in 
2  yrs.  3  mos.,  at  6%. 

2  yrs.  3  mos.  =  2.25  yrs. 

Interest  for  1  yr.  =  ^^^^^'^^  =  $ 583.40. 
^  2.25 

Interest  on  $1  for  1  yr.,  at  6%  =  $0.06. 

Hence,  principal  required  =  — — —  of  $1  =  $9723.33.  Ans. 

62.  Find  the  principal  that  will  produce  $  750  interest  in  3  yrs. 
8  raos.,  at  5%. 

3  yrs.  8  mos.  =  3|  yrs. 

Interest  for  1  yr.  =  ^^  =  $  204.5455. 
3f 

Interest  on  $1  for  1  yr.,  at  5%  =  $0.05. 

Hence,  principal  required  = ^ of  $  1  =  $4090.91.  Ans. 

63.  Find  the  principal  that  will  amount  to  $840  in  3  yrs.,  at  4%. 
If  the  principal  be  represented  by  100,  the  interest  will  be  repre- 
sented by  3  X  4  =  12,  and   the   amount  will   be   represented 
by  112.     Hence,  the  principal  =  iff  of  $  840  =  $  750.  Ans. 

64.  Find  the  principal  that  will  amount  to  $901.1384  in  2  yrs. 
6  mos.,  at  41%. 

2  yrs.  6  mos.  =  2|  yrs. 

If. the  principal  be  represented  by  100,  the  interest  will  be  repre- 
sented by  21  X  4|-  =  10y\,  and  the  amount  will  be  represented 
by  110^. 

Hence,  the  principal  =  -^^  of  $  901.1384  ==  $  816.896.  Ans. 

65.  Find  the  principal  that  will  amount  to  $  6000  in  21  dys.,  at 

21  days.  =  -^^  yr- 

If  the  principal  be  represented  by  100,  the  interest  will  be  repre- 
sented by  -^-1^X5  =  -^j,  and  the  amount  will  be  represented 
by  lOOx^^. 

Hence,  the  principal  =  -^^  of  $  6000  -  $  5982.55.  Ans. 
^        ^  lOO/j      ^ 


324  ARITHMETIC. 


66.  Find  the  principal  that  will  amount  to  $  297.60  in  8  mos., 
at  6%. 

8  mos.  =  I  yr. 

If  the  principal  be  represented  by  100,  the  interest  will  be  repre- 
sented by  §  X  6  =  4,  and  the  amount  will  be  represented 
by  104. 

Hence,  the  principal  =  |^f  of  $297.60  =  $286.15.  Ans. 

67.  Find  the  principal  that  will  amount  to  $6378.75  in  1  yr.  1  mo,, 
at  5%. 

1  yr.  1  mo.  =  l^V  yr- 

If  the  principal  be  represented  by  100,  the  interest  will  be  rep- 
resented by  ly^^  X  5  =  5y\,  and  the  amount  will  be  represented 
by  105^. 

Hence,  the  principal  =  -^^  of  $6378.75  =  $6050.99.  Am. 

68.  Find  the  principal  that  will  amount  to  $21,047.95  in  1  yr. 
7  moa.  21  dys.,  at  4|%. 

1  yr.  7  mos.  21  dys.  =  1//^  yr. 

If  the  principal  be  represented  by  100,  the  interest  will  be  rep- 
resented by  l^y^  X  4^  =  7|^,  and  the  amount  will  be  repre- 
sented by  107f^. 

Hence,  the  principal  =  -^^  of  $21,047.95  =  $  19,600.  Ans. 

69.  Find  the  principal   that  will   amount  to  $185.09  in  2  yrs. 
3  mos.  18  dys.,  at  5%. 

2  yrs.  3  mos.  18  dys.  =  2^^  yrs. 

If  the  principal  be  represented  by  100,  the  interest  will  be  rep- 
resented by  2i^x  5  =  11-^jj,  and  the  amount  will  be  repre- 
sented by  llli'V- 

Hence,  the  principal  =  -^-  of  $  185.09  =  $  166.  Ans. 
11  If  J 

70.  Find  the  principal  that  will  amount  to  $659.40  in  2  yrs. 
11  mos.  15  dys.,  at  6%. 


teachers'  edition.  325 

2  yrs.  11  raos.  15  dys.  =  2|f  yrs. 

If  the  principal  be  represented  by  100,  the  interest  will  be  rep- 
resented by  2f  f  X  6  =  17f ,  and  the  amount  will  be  represented 
by  117|. 

Hence,  the  principal  =  ^^^  of  $659.40  =  |560.  Ans. 

71.    Find  the  principal  that  will  amount  to  1 9437.516  in  2  yrs. 
7mos.  24  dys.,  at4|%. 

2  yrs.  7  mos.  24  dys.  =  2^f  yrs. 

If  the  principal  be  represented  by  100,  the  interest  will  be  rep- 
resented by  2|f  x4i  =  llf^,  and  the  amount  will  be  repre- 
sented by  lllf^. 

Hence,  the  principal  =  -^^  of  $9437.516  =  $8432.  Ans. 
^        ^         lllf^ 


72.   Find  the  principal  that  will  amount  to  $10,266.60  in  3  yrs. 
5  mos.,  at  6%. 

3  yrs.  5  mos.  =  3i^7^  yrs. 

If  the  principal  be  represented  by  100,  the  interest  will  be  repre- 
sented by  3y\  X  6  =  20^,  and  the  amount  will  be  represented 
by  120i. 

Hence,  the  principal  =  ^^  of  $10,266.60  =  $8520.  Ans. 


73.   What  is  the  interest  of  $195  for  2  yrs.  2  mos.  2  dys.,  at  61  % 

2  yrs.  =0.12  $195 

2  raos.  =  0.01  X0.14H 

2  dys.  =0000^  $27.53.  ^m. 

12)0.130^ 
0.01  Of  ^ 


74.   At  what  rate  per  cent  will  $1025.20  produce  $25.72  in  4  mos. 
9  dys.  ? 


32G  ARITHMETIC. 


4  mos.  9  days.  =  0.3583  yr. 

Interest  on  $1025.20  for  0.3583  yr.  is  $25.72  ; 

on  $1025.20  for  1  yr.  is  ^^^  ; 
•^         0.3583 

on  $  1  for  1  yr.  is 125^72 _  ^^^^  nearly. 

^  ^         0.3583  X  1025.20     ^  ^ 

Hence,  the  rate  required  is  7%,  nearly.  -4ns. 

75.  The  principal  is  $653 ;  the  interest  $  5.52 ;  the  rate  8%.    Find 
the  time. 

Interest  on  $653,  at  8%,  for  1  yr.  is  0.08  of  $653  =  $52.24. 

$5  52 
Therefore,  the  number  of  years  will  be  "^^ —  =  0.1056. 
^  52.24 

0.1056  yr.  =  1  mo.  8  dys.  Ans. 

76.  Find  the  amount  of  $520  for  2  mos.  3  dys.,  at  4|%. 

2  mos.  =  0.01  $  520 

^  3  dys.  =  0.0005  x  0.0078| 

4)0.0105  $4.10 

0.00261-  520.00 


0.0078f  $524.10.  Ans. 

77.   What  sum  bearing  interest  at  4^%  will  yield   an   annual 
income  of  $  1000 '? 

Interest  on  $1  for  1  yr.,  at  4J%  =  $0,045. 

Hence,  principal  required  =  ^^  of  $1  =$22,222.22.  Ans. 


78.  Kow  long  will  it  take  $4000  to  produce  $625  interest,  at  5J%? 

Interest  on  $4000,  at  5^%,  for  1  yr.  is  0.055  of  $4000  =  |220. 

Therefore,  the  number  of  years  will  be  |-^  =  2.841. 
^  $220 

And  2.841  yrs.  =  2  yrs.  10  mos.  3  dys.  Ans. 

79.  At  what  rate  per  cent  will  $3000  produce  $250  interest  in 
1  yr.  2  mos.  24  dys.  ? 


teachers'  edition.  327 


1  yr.  2  mos.  24  dys.  =  1/,^  yrs. 
Interest  on  1 3000  for  I3V  yrs.  is  $250  ; 

on  1 3000  for  lyr.  is  i;^; 

1-3(7 

Hence,  the  rate  required  is  6f  f  %  Ans. 


80.   Find  the  interest  of  $1721.84  from  April  1  to  Nov.  12  at  A\%. 

mos.  dys. 

11  12 

4  1 


7  11 


7  mos.  =0.035  $1721.84 

11  dys.  =  O.OOlf  X  0.027f 

4)0.036f  $47.57.  Am. 

0.009/^ 

0.027f 

81.  How  long  must  $  3904.92  be  on  interest  to  amount  to  $  4568.76, 
at  5%. 

The  interest  =  $4568.76  -  $3904.92  =  $663.84. 
Interest  on   $3904.92,   at  5%,  for  1  yr.   is  0.05  of  $3904.92 
=  $195,246. 

Therefore,  the  number  of  years  will  be  J^     '  '       =  3.4. 

<|)  iyo.^4o 

And  3.4  yrs.  =  3  yrs.  4  mos.  24  dys.  Ans. 

82.  Find  the  interest  of  $137.60  from  July  3  to  Dec.  12,  at  7t^o%- 


mos.                     dys. 

12                12 
7                 3 

$137.60 
X  0.073 
12)$  10.04480.  Int.  fori  yr. 

5                 9 
=  5.3  mos. 

$0.837075.  Int.  for  1  mo. 
X  5.3 

$4.44.  Ans. 


328  ARITHMETIC. 


83.    Find  the  interest  of  1 680.20,  at  7^%  for  73  dys.,  reckoning 
365  dye.  for  a  year. 

73dy8.  =  ^/^yr.=iyr. 

$680.20 
X  0.07^ 


6)$51.0150 
$10.20.  Ans. 


Exercise  LXXIL 

Find  day  of  maturity,  the  time  to  run,  the  discount,  and  proceeds  of 
the  following  notes : 

1.  $750.  New  York,  Jan.  1,  1881. 
Four  months  from  date  I  promise  to  pay  to  the  order  of  James 

Fay  seven  hundred  and  fifty  dollars,  value  received. 

Discounted  at  7%,  Jan.  12.  John  Peay. 

The  note  becomes  due  4  mos.  from  Jan.  1  =  May  ^/^. 

The  time  to  run  is  19  dys.  in  January,  28  in  February,  31  in 

March,  30  in  April,  and  4  in  May  =  112  dys. 
The  discount  is  the  interest  on  $750,  for  112  dys.,  at  7%. 
Therefore,  the  discount  is  0.021^  of  $  750  =  $  16.33. 
And  the  proceeds  is  $  750  -  $  16.33  =  $  733.67.  Ans. 

2.  $4325.50.  Boston,  Jan.  3,  1881. 
Sixty  days  from  date  I  promise  to  pay  to  James  Finn,  or  order, 

four    thousand   three   hundred  twenty-five  and  {)j%  dollars,  value 
received. 

Discounted  at  6|%,  Jan.  6.  George  Bellows. 

Counting  60  days,  from  Jan.  3,  there  are  28  in  January,  28  in 

February,  and  4  in  March. 
Therefore,  the  note  becomes  due  March  */^. 
The  time  to  run  is  25  dys.  in  January,  28  in  February,  and  7  in 

March  =  60  dys. 
The  discount  is  the  interest  on  $4325.50,  for  60  dys.,  at  6J%. 
Therefore,  the  discount  is  O.OlOf  of  $4325.50  =  $46.86. 
And  the  proceeds  is  $4325.50  -  $46.86  =  $4278.64.  Ans.    • 


teachers'  edition.  329 

3.  $  1340.70.  Richmond,  Va.,  Jan.  6,  1881. 
Ninety  days  from  date  I  promise  to  pay  to  the  order  of  Peter 

Bright  thirteen  hundred  forty  and  j^-^q  dollars,  value  received. 
Discounted  at  7%,  Jan.  26.  Geokge  Weight. 

Counting  90  dys.,  from  Jan.  6,  there  are  25  in  January,  28  in 

February,  31  in  March,  and  6  in  April. 
Therefore,  the  note  becomes  due  April  "/g. 
The  time  to  run  is  5  dys.  in  January,  28  in  February,  31  in 

March,  and  9  in  April  =  73  dys. 
The  discount  is  the  interest  on  $1340.70,  for  73  dys.,  at  7%. 
Therefore,  the  discount  is  0.014/^  of  $1340.70  =  $19.03. 
And  the  proceeds  is  $  1340.70  -  $  19.03  =  $  1321.67.  Ans. 

4.  $1456.30.  Chaeleston,  S.C,  Jan.  19,  1881. 
Three  months  after  date  I  promise  to  pay  to  the  order  of  John 

George  fourteen  hundred  fifty-six  and  j"^^^  dollars,  value  received. 
Discounted  at  5%,  Feb.  1.  John  Waldoef. 

The  note  becomes  due  3  mos.  from  Jan.  19  =  April  ^^l22- 
The  time  to  run  is  2  mos.  21  dys. 

The  discount  is  the  interest  on  $  1456.30,  for  2  mos.  21  dys.,  at  5%. 
Therefore,  the  discount  is  0.01125  of  $1456.30  =  $16.38. 
And  the  proceeds  is  $  1456.30  -  $  16.38  =  $  1439.92.  Ans. 

5.  $4550.36.  Deteoit,  Mich.,  Feb.  2,  1881. 
Four   months  after  date  I  promise  to  pay  to  the  order  of  John 

Callender  four   thousand  five  hundred  fifty  and  j^^-^  dollars,  value 
received. 

Discounted  at  5|%,  Feb.  16.  James  Baeton. 

The  note  becomes  due  4  mos.  from  Feb.  2  =  June  2/,. 

The  time  to  run  is  3  mos.  19  dys. 

The  discount  is  the   interest   on  $4550.36,  for  3  mos.  19  dys., 

at  5J  %. 
Therefore,  the  discount  is  0.0166if  of  $4550.30  =  $75.78. 
And  the  proceeds  is  $4550.36  -  $  75.78  =  $  4474.58. 

6.  $5000.  Chicago,  111.,  Oct.  4,  1880. 
Six  months  after  date  I  promise  to  pay  to  John  Adams,  or  order, 

five  thousand  dollars,  value  received,  with  interest  at  seven  per  cent. 
Discounted  at  8  %,  Dec.  31.  William  Dunn. 


330  ARITHMETIC. 


Interest  on  note  for  6  mos.  3  dys.  =  $  177.92. 

Amount  of  note  $  5000  +  $  177.92  =  1 5177.92. 

Day  of  maturity,  April  ^/y. 

Time  to  run,  3  mos.  6  dys. 

Discount  on  $5177.92,  at  8%,  for  3  mos.  6  dys.  =  $110.46. 

Proceeds  is  $5177.92 -$110.46  =  $5067.46.  Am. 

7.  $4760.  Milwaukee,  Wis.,  Jan.  1,  1881. 
Ninety  days  after  date  I  promise  to  pay  to  the  order  of  James 

Pike  four  thousand  seven  hundred  and  sixty  dollars,  value  received. 
Discounted  at  11%,  Feb.  15.  William  Clement. 

Counting  90  dys.  from  Jan.  1,  there  are  30  in  January,  28  in 

February,  31  in  March,  and  1  in  April. 
Therefore,  the  note  becomes  due  April  ^/^. 
The  time  to  run  is  13  dys.  in  February,  31  in  March,  and  4  in 

April  =  48  dys. 
The  discount  is  the  interest  on  $4760  for  48  dys.,  at  7^%. 
Therefore,  the  discount  is  0.01  of  $4760  =  $47.60. 
And  the  proceeds  is  $4760  -  $47.60  =  $4712.40.  Ans. 

8.  $2017.85.  Kansas  City,  Mo.,  Jan.  14,  1881. 
Three  months  after  date  I  promise  to  pay  to  the  order  of  John 

Brown  two  thousand  seventeen  and  ^^j^  dollars,  value  received. 
Discounted  at  10%,  Mar.  1.  Timothy  Bruce. 

The  note  becomes  due  3  mos.  from  Jan.  14  =  April  ^Vi?. 

The  time  to  run  is  1  mo.  16  dys. 

The  discount  is  the  interest  on  $  2017.85  for  1  mo.  16  dys.,  at 

10%. 
Therefore,  the  discount  is  0.012|  of  $2017.85  =  $25.78. 
And  the  proceeds  is  $  2017.85  -  $  25.78  =  $  1992.07.  Ans. 

9.  $652.45.  Concord,  N.H.,  Jan.  25,  1881. 
Five  months  after  date  I  promise  to  pay  to  the  order  of  Charles 

Barrett  six  hundred  fifty-two  and  ^^  dollars,  value  received,  with 
interest  at  six  per  cent. 
Discounted  at  4^%,  Mar.  15.  William  Kimball. 

Interest  on  note  for  5  mos.  3  dys.  =  $16.64. 
Amount  of  note  $  652.45  +  $  16.64  =  $  669.09. 


teachers'  edition.  331 

Day  of  maturity,  June  ^^/gg- 

Time  to  run,  3  mos.  13  dys. 

Discount  on  |669.09,  at  4i%,  for  3  mos.  13  dys.  =$8.61. 

Proceeds  is  1 669.09  - 1 8.61  =  $660.48.  Ans. 


10.  $9040.  Baltimore,  Md.,  Jan.  19,  1881. 
Sixty  days  from  date  I  promise  to  pay  to  the  order  of  Charles  Car- 
roll nine  thousand  and  forty  dollars,  value  received. 

Discounted  at  5|%,  Feb.  16.  James  Monroe. 

Counting  60  dys.,  from  Jan.  19,  there  are  12  in  January,  28  in 

Febniary,  and  20  in  March. 
Therefore,  the  note  becomes  due  March  ^"/gs- 
The  time  to  run  is  12  dys.  in  February  and  23  in  March  =  35 

dys. 
The  discount  is  the  interest  on  $9040,  for  35  dys.,  at  5|%. 
Therefore,  the  discount  is  O.OOyVV  of  $9040  =  $48.34. 
And  the  proceeds  is  $9040  -  $48.34  =  $8991.66.  Ans. 

11.  $215.  Augusta,  Me.,  Jan.  28,  1881. 
Thirty  days  after  date  I  promise  to  pay  to  the  order  of  James 

Fogg  two  hundred  and  fifteen  dollars,  value  received. 
Discounted  at  6%,  Feb.  3.  John  Moses. 

Counting  30  dys.,  from  Jan.  28,  there  are  3  in  January  and  27 

in  February. 
Therefore,  the  note  becomes  due  ^^^-  '^Viiarch  2- 
The  time  to  run  is  25  dys.  in  February  and  2  in  March  =  27  dys. 
The  discount  is  the  interest  on  $215,  for  27  dys.,  at  6%. 
Therefore,  the  discount  is  0.0045  of  $215  =  $0.97. 
And  the  proceeds  is  $215  -  0.97  =  $  214.03.  Ans. 

12.    Find  the  face  of  a  note  at  90  dys.  that  will  realize  $850  when 
discounted  at  7%. 

The  discount  on  $1  for  93  dys.  is  $0.0180^. 
Proceeds  on  $1  is  $1  -  $0.0180f  =  $0.9819^. 

Therefore,  the  face  required  for  $850  is  pr|^  =  $8^5.65.  Ans. 

0,00  Lv-^ 


332  ARITHMETIC. 


13.  Find  the  face  of  a  note  at  4  mos.  that  will  realize  $  1600  when 
discounted  at  5^  %. 

The  discount  on  1 1  for  4  mos.  3  dys.  is  |0.0187ii 
Proceeds  on  1 1  is  $  1  - 1 0.0187H  =  1 0.9812^^. 

Therefore,  the  face  required  for  $1600  is    ^^^^^   =  $1630.64. 

0.98 12j  J 

14.  Find  the  face  of  a  note  at  30  dys.  that  will  realize  $  1200  when 
discounted  at  6^%. 

The  discount  on  $1  for  33  dys.  is  $0.0059^?^. 
Proceeds  on  $1  is  $1  - 10.0059/^  =  $0.9940^5^. 

Therefore,  the  face  required  for  $  1200  is  tr|^|^  =  $  1207.19. 

15.  Find  the  face  of  a  note  at  60  dys.  that  will  realize  $  4000  when 
discounted  at  8  %. 

The  discount  on  $1  for  63  dys.  is  $0.0140. 
Proceeds  on  $  1  is  $  1  -  $0,014  =  $0,986. 

Therefore,  the  face  required  for  $4000  is  ^^^^  =  $  4056.80.  Ans. 
^  0.986 

16.  Find  the  face  of  a  note  at  2  mos.  that  will  realize  $4500  when 
discounted  at  7/^^%. 

The  discount  on  $  1  for  63  dys.  is  $0.012775. 
Proceeds  on  $1  is  $1  -  $0.012775  =  $0.987225. 

Therefore,  the  face  required  for  $4500  is    ^^^^    =  $4558.23. 
^  0.987225     ^ 

17.  Find  the  face  of  a  note  at  3  mos.  that  will  realize  $1100  when 
discounted  at  7%. 

The  discount  on  $1  for  3  mos.  3  dys.  is  $0.01808f 
Proceeds  on  $1  is  $1  -  $0.018082  =.$0.98191|. 

Therefore,  the  face  required  for  $  1 100  =    ^^^^^    =  $  1 220.26. 
^  ^  0.98191^     * 

18.  Find  the  present  worth  of  $500  due  in  11  mos.  at  5%. 

If  100  be  taken  to  represent  the  present  worth,  tlie  discount  will 
be  represented  by  -J-i  X  5  =»  4^5^. 


teachers'  edition.  333 

The  given  sum  will  be  represented  by  104y''Tj. 

Hence,  the  present  worth  is  -^^  of  $500  =  $478.09.  Ans. 

19.  Find  the  present  worth  and  discount  of  1 3334.62  due  in  2  yrs., 
at4r/o. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  2x4.^==  9. 
The  given  sum  will  be  represented  by  109. 
Hence,  the  present  worth  is  |fg  of  $3334.62  =  $3059.28;  and 

the  discount  is  $3334.62  -$3059.28  =  $275.34.  Ans. 

20.  Find  the  present  worth  and  discount  of  $4261.33  due  at  the 
end  of  1  yr.  6  mos.,  at  6%. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  1^  X  6  =  9. 
The  given  sum  will  be  represented  by  109. 
Hence,  the  present  worth  is  |^f  of  $4261.33  =  $3909.48. 
The  discount  is  $4261.33  -  $3909.48  =  $351.85.  Ans. 

21.  Find  the  present  worth  and  discount  of  $2416.60  due  in  7 
mos.,  at  5  %. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  xV  X  5  =  2\^. 

The  given  sum  will  be  represented  by  102|^, 

100 

Hence,  the  present  worth  is  of  $2416.50  =  $2348.02,  and 

102|^ 

the  discount  is  $2416.50  -  $2348.02  =  $68.48.  Ans. 

22.  Find  the  present  worth  of  $678.40  due  in  16  mos.,  at  4|%. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  1^x4^  =  6. 
The  given  sum  will  be  represented  by  106. 
Hence,  the  present  worth  is  \%^  of  $678.40  =  $640.  Ans. 

23.  Find  the  present  worth  and  discount  of  $574.17  due  in  2  yrs, 
3  mos.,  at  5|%. 


334  ARITHMETIC. 


If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  2\  x  5}  =12. 
The  given  sum  will  be  represented  by  112, 
Hence,  the  present  worth  is  }^g  of  1 574.17  =  $512.65;  and  the 

discount  is  $574.17  -  $512.65  =  $61.52.  Ans. 


24.   Find  the  present  worth  and  discount  of  $625.13  due  in  8  mos., 
at  7^%. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  |  X  7^^  =  4||. 
The  given  sum  will  be  represented  by  104|f. 

Hence,  the  present  worth  is  — —  of   $625.13  =  $596.12;    and 
104|| 

the  discount  is  $625.13  -  $596.12  =  $29.01.  Ans. 


25.   Find  the  present  worth  and  discount  of  $  715.20  due  in  1  yr. 
4  mos.,  at  3^%. 

If  100  be  taken  to  represent  the  present  worth,  the  discount  will 

be  represented  by  H  X  3^  =  4f. 
The  given  sum  will  be  represented  by  104|. 

Hence,  the  present  worth  is  ^^  of  $  715.20  =  $  683.31 ;  and  the 
discount  is  $715.20 -$683.31  =  $31.89.  Ans. 


26.  If  I  buy  goods  Jan.  10,  at  30  dys.,  for  $218;  Feb.  5,  at  60 
dys.,  for  $421 ;  and  pay  Feb.  10,  $200,  March  17,  $50,  what  is  due 
April  25,  interest  at  6% ? 

$218  +  2  mos.  16  dys.  interest  =  $220.80 

$421  + 19  dys.  interest  -  422.33 


$200  +  2  mos.  15  dys.  interest  =  $202.50 
$50    +  1  mo.     8  dye.  interest  =      50.32 


$643.13 
$252.82 


Balance  due,  $390.31.  iltu. 


teachers'  edition.  335 

27.  A  note  for  1 618.75,  dated  April  17,  1880,  payable  on  demand, 
bears  the  following  endorsements:  June  5,  $126.50;  Aug.  20, 
$137.25  ;  Nov.  17,  $210.  What  is  due  Jan.  1,  1881,  reckoning  in- 
terest at  6%  ? 

Amount  of  $  618.75  for  8  mos.  14  dys.  =  $  644.94 

Amount  of  $126.50  for  6  mos.  26  dys.  =  $  130  84 
Amount  of  $  137.25  for  4  mos.  11  dys.  =  140.25 
Amount  of  $210      for  1  mo.    14  dys.  =    211.54 

$482.63 

Balance  due,  $162.31.  ^ns. 

28.  A  note  for  $  1000,  dated  April  1,  1880,  payable  on  demand, 
with  interest  at  7%,  bears  the  following  endorsements  :  May  6,  $200; 
July  5,  $225.37;  Oct.  18,  $322.     What  is  due  Jan.  1,  1881? 

Amount  of  $  1000  for  9  mos.  =  $  1052.50 

Amount  of  $  200  for  7  mos.  25  dys.  =  $  209.14 
Amount  of  $  225.37  for  5  mos.  26  dys.  =  233.08 
Amount  of  $322  for  2  mos.  13  dys.      =    326.57 

$768.79 

Balance  due,  $283,71.  Ans. 


29.  A  note  for  $835.25,  dated  July  1,  1880,  payable  on  demand, 
with  interest  at  6^%,  bears  the  following  endorsements :  Aug.  20, 
$157.50;  Sept.  21,  $180.25;  Oct.  5,  $200;  Dec.  1,  $80.  What  is 
due  Jan.  1,1881? 

Amount  of$835.25  for  6  mos.  =  $862.39 

Amount  of  $157.50  for  4  mos.  11  dys.  =  $161.23 
Amount  of  $  180.25  for  3  mos.  10  dys.  =  183.49 
Amount  of  $  200  for  2  mos.  26  dys.  =  203.11 
Amount  of  $80  for  1  mo.  =      80.43 

$628.26 
Balance  due,  $234.13.  ^m. 


336  ARITHMETIC. 


30.  A  note  of  $2000,  dated  Jan.  22,  1880,  and  drawing  interest  at 
6%,  had  payments  endorsed  upon  it  as  follows:  May  20,  1880,  $100; 
July  20,  1880,  $325;  Nov.  2.  1880,  $20;  Dec.  23,  1880,  $125.  Find 
the  balance  due  March  1,  1881. 

$2000 
0.019f 

$39.33  Ist  interest. 
2000.00 


yr- 

mo8. 

dyg. 

1880 

5 

26 

1880 

1 

22 

3        28    0.0191 


1880 

1880  5        20 


$2039.33 
c  100.  100.00  Ist  payment. 

H        20  $1939.33  2d  principal. 

0.01 


Q  Q-.  $  19.40  2d  interest. 

1939.33 


$1958.73 
1 325.  325.00  2d  payment. 

1880        11  2  $1633.73  3d  principal. 

1880  7        20  0.017 


12    0.017  $20      $27.78  3d  interest. 


$20. 


$1633.73  3d  principal. 
0.0085 


j2        23  ^'^^     $13.88  4th  interest. 


1880 

1880        11  2 


27.78  3d  interest. 
1633.73 


21    0.0085 


$1675.39 

145.00  3d  &  4th  payments. 


$570.  $  1530.39  4th  principal. 


1881  3  1 


O.OIH 


1880        12        23  $17.34  5th  interest. 

1530.39 


8    O.Olli 


$1547.73  Am, 


teachers'  edition.  337 

31.  A  note  of  $  1662.50,  dated  Jan.  15,  1880,  and  drawing  interest 
at  6|%,  had  payments  endorsed  upon  it  as  follows:  April  30,  1880, 
1 25;  June  24,  1880,  |25;  Sept.  2,  1880,  |625;  Jan.  31,  1881,  $700. 
Find  the  balance  due  May  12,  1881. 


jr. 

mos.        dys. 

$1662.50 

1880 

4        30 

0.0189^j 

1880 

1        15 

$25      $31.52  1st  interest. 

3        15 

0.0189^^ 

$  1662.50  1st  principal. 
0.0097^ 

$25. 

$25      $16.21  2d  interest. 

1880 

6        24 

31.52  1st  interest. 

1880 

4        30 
1        24 

0.0097^ 

1662.50 

$1710.23 

50.00  1st  &  2d  payments. 

$1660.23  2d  principal. 

$25. 

0.012^ 

1880 

9          2 

$  20.38  3d  interest. 

1880 

6        24 

1660.23 

2          8 

0.012^y 

$1680.61 

625.00  3d  payment. 

$625. 

$1055.61 

0.026f| 

1881 

1880 

1        31 

9          2 

$28.40  4thinierest. 
1055.61 

4        29 

0.026fi 

$1084.01 

700.00  4th  payment. 

$700. 

$  384.01  4th  principal. 

1881 

5        12 

0.018H 

1881 

1        31 

$7.00 

3        11 

0.018^1 

384.01 
$391.01  Am. 

338  AEITHMETIC. 


32.  A  note  of  |4560,  dated  Jan.  22,  1879,  and  drawing  interest  at 
7%,  had  payments  endorsed  upon  it  as  follows :  Jan.  10,  1880,  $2000 ; 
Aug.  31,  1880,  $500;  Jan.  15,  1881,  $1200;  March  4,  1881,  $860. 
Find  the  balance  due  June  15,  1881. 


yr. 

mos.          iya. 

$4560 

1880 

1        10 

0.067f 

1879 

1        22 

$308.56  Ist  interest. 

11        18 

0.067f 

4560.00 

$4868.56 

$2000. 

2000.00  1st  payment. 
$2868.56  2d  principal. 

1880 

8        31 

0.0449^ 

1880 

1        10 

$128.85  2d  interest. 

7        21 

0.0449^ 

2868.56 

$2997.41 

500.00  2d  payment. 

$500. 

$2497.41  3d  principal. 

1881 

1        15 

0.026^^ 

1880 

8        31 

$65.07  3d  interest. 

4        14 

0.026TJff 

2497.41 

$2562.48 
1200.00  3d  payment. 

$1200. 

$1362.48  4th  principal. 

1881 

3          4 

0.009Jf 

1881 

1         15 

0.009H 

$12.98  4th  interest. 

1        19 

1362.48 

$1375.46 

860.00  4lh  payment. 

$860. 

$515.46  5th  principal. 

1881 

6        15 

0.018IJ 

1881 

3          4 

$10.12  5th  interest. 

3        11 

0.01811 

$515.46 
$525.58  Ans. 

teachers'  edition.  339 


33.  A  note  of  $  785.50,  dated  Jan.  30,  1879,  and  drawing  interest 
at  5%,  had  payments  endorsed  upon  it  as  follows:  July  17,  1879, 
1 100;  Jan.  29,  1880,  $100;  Dec.  31, 1880,  |20;  Mar.  16, 1881,  |300; 
June  14,  1881,  |50.     Find  the  balance  due  July  23,  1881. 


1879 
1879 

mos.    dys. 
7     17 

1    30 

5    17 

0.023f 

1100. 

1880 
1879 

1    29 

7    17 

$785.50 
0.023ff 

$18.22  1st  interest. 

785.50 


$803.72 

100.00  1st  payment. 
$703.72  2d  principal. 
0.02| 
$  18.77  2d  interest. 
6         12    0.02f  703.72 

$722.49 
$  100  100.00  2d  payment. 

1880        12        31  $622  49  3d  principal. 


1880  1        29 


11    2 

$20. 

1881 

3    16 

1880 

12   31 

2    15 

$300. 

1881 

6   14 

1881 

3    16 

2    28 

$50. 

1881 

7   23 

1881 

6   14 

0.046^ 
$ 20     $28.71  3d  interest. 
^■^"^^h  $622.49  3d  principal. 

0.010, 


'a:2 


$6.48  4th  interest. 
28.71  3d  interest. 
622.49 


$657.68 
■      ^^  320.00  3d  &  4th  payments. 

$337.68  4th  principal. 
OOlf 

$4.13  5th  interest. 
337.68 


0.011  $341.81 

50.00  5th  payment. 
$291.81  5th  principal. 
0.005t\ 
$1.58  6th  interest. 
291.81 
1  9    0.005x\  $293.39  Ans, 


340  ARITHMETIC. 


34.  A  note  of  $300.25,  dated  Aug.  4,  1879,  and  drawing  interest 
at  6^%,  had  payments  endorsed  upon  it  as  follows:  Oct.  14,  1879, 
$100;  July  21,  1880,  $100;  Oct.  11,  1880,  $50;  Jan.  18,  1881,  $50. 
Find  the  amount  due  July  22,  1881. 


mos. 


1879        10        14 
1879  8  4 


dy».  $300.25 

0.012ff 


10    0.012f| 


$100. 

1880    7   21 
1879   10   14 


9  7    0.049H 


$100. 

1880        10        14 

1880  7        21 


2        23    0.014^^ 


$50. 


1881    1   18 
1880   10   11 


7    0.017U 


$50. 

1881         1       22 
1881  1        18 


6         4    0.033f 


$3.79  Ist  interest. 
300.25 

$304.04 
100.00  Ist  payment. 

$204.04  2d  principal. 
0.049H 

$10.14  2d  interest. 
204.04 

$214.18 
100.00  2d  payment. 

$114.18  3d  principal. 
0.014H 

$1.71  3d  interest. 
114.18 

$115.89 

50.00  3d  payment. 

$65.89  4th  principal. 
0.017H 

$1.15  4th  interest. 
65.89 

$67.04 
50.00  4th  payment. 

$17.04  5th  principal. 
0.033f 

$0.57  5th  interest. 
17.04 

$17.61  Am. 


TEACHERS     EDITION. 


341 


35.   Find     the     amount      of 

37. 

Find  the  compound  inter- 

$356.25   in  4  yrs.,  at  5%,  com- 

est of  $800  in  3  yrs.  9  mos.,  at 

pound  interest. 

6%. 

$356.25 

$800.00 

0.05 

0.06 

$17.81 

$48.00 

365.25 

800.00 

$374.06 

$848.00 

0.05 

0.06 

$18.70 

$50.88 

374.06 

848.00 

$392.76 

$898.88 

0.05 

0.06 

$19.64 

$53.93 

392.76 

898.88 

$412.40 

$952.81 

0.05 

0.04^ 

$20.62 

$42.88 

$412.40 

952.81 

$433.02  Ans. 

$995.69 
800.00 
$195.69  Ans. 

36.   Find     the     amount     of 

$637.50  in  2  yrs.  6  mos.,  at  4%, 

compound  interest. 

38. 

Find  the  compound  inter- 

$637.50 

est  of 
5%. 

$39.35  in  1  yr.  9  mos.,  at 

0.04 

$25.50 

$39.35 

637.50 

0.05 

$663.00 

$1.97 

0.04 

39.35 

$26.52 

$41.32 

663.00 

0.03f 

$689.52 

$1.55 

0.02 

41.32 

$13.79 

$42.87 

689.52 

39.35 

$703.31  Ans. 

$3.52  Ans. 

342  ARITHMETIC. 


39.   Find  the  compound  interest  of  $300  in  2  yrs.,  at  4%,  interest 
being  payable  semi-annually. 

$300  $312.12 

0.02  0.02 


$6.00  $6.24 

300.00  312.12 


$306.00  $318.36 

0.02  0.02 


$6.12  $6.37 

306.00  318.36 


$312.12  $324.73 

300.00 


$24.73  Am. 

40.   Find  the  compound  interest  of  $525  in  1  yr.  6  mos.,  at  5%, 
interest  being  payable  semi-annually. 

$525  $544.93 

O.OIJ  0.01^ 

$6.56  $6.81 

525.00  544.93 


$531.56  $551.74 

O.OIJ  0.01^ 

$0.64  $6.90 

531.56  551.74 


$538.20  $558.64 

0.01^  O.Oli 

$6.73  $6.98 

538.20  558.64 


$644.93  $565.62 

525.00 


$40.62  Ans. 


TEACHERS     EDITION. 


343 


41.    Find  the  compound  interest  of  $10,000  in  6  mos.,  at  6%,  in- 
terest being  paid  monthly. 


$10000 
0.005 

$50.00 
10000.00 

$10050.00 
0.005 


$  10150.75 
0.005 


$50.75 
10150.75 

$10201.50 
0.005 


$50.25 
10050.00 

$10100.25 
0.005 

$50.50 
10100.25 

$10150.75 


$51.01 
10201.50 

$10252.51 
0.005 


'      $51.26 
10252.51 

$10303.77 
10000.00. 


$303.77  Ans. 

42.  What  principal  will  amount  to  $137.81  in  2  yrs.,  at  5%  com- 
pound interest? 

The  amount  of  $  1  for  2  yrs.  at  5  %  is  $  1  X  l.OS^  =  $  1.1025. 
The  principal  is  $137.81  -^  1.1025  =  $125.  Ans. 

43.  What  principal  will  amount  to  $1860.96  in  3  yrs.,  at  6%  com- 
pound interest? 

The  amount  of  $1  for  3  yrs.  at  6%  is  $1  X  l.Oe^  =  $1.191016. 
The  principal  is  $1860.96  ^  1.191016  =  $1562.50.  Ans. 


44.   What  principal  will  amount  to  $1500  in  1  yr.,  at  4%  com- 
pound interest,  payable  quarterly? 

The  amount  of  $1  for  1  yr.  at  4%,  payable  quarterly,  is 

$1x1.013  =  $1.04060401. 
The  principal  is  $  1500  h-  1.04060401  =  $1441.47.  Ans. 


344  ARITHMETIC, 


45.  "What  principal  will  produce  $100  in  1  yr.  6  mos,,  at  6%  com- 
pound interest,  payable  semi-annually  ? 

The  amount  of  $1  for  1  yr.  6  mos.,  payable  Bemi-annually,  is 

$1x1.033  =  $1.092727. 
The  interest  is  $1.092727  -  $1  =  $0.092727. 
The  principal  is  $  100  +  0.092727  =  $  1078.43.  Ans. 

46.  Find  the  interest  due  May  19,  1881,  on  a  note  dated  Dec.  26, 
1877,  for  $1224.60,  with  interest  payable  annually  at  5%,  when  no 
interest  has  been  paid. 


yrs.       mos.     dys.  $1224.60 

1881   5  19 
1877  12  26 


0.203f 

6)249.61 
3      4    23    0.203^  ^^qq 


$208.01  interest  for  3  yrs.  4  mos.  23  dys. 


yrs.   mos.  dys.                                           $1224.60 

2     4  23                                      0.05 

1     4  23 

4 


23  $61.23  annual  interest. 

0.05 


4     2       9  =  4f^  yrs. 


$3.0615  interest  on  annual  interest. 

$12.83 
208.01 


$220.84  total  interest.  Am. 

47.  Find  the  amount  due  May  27,  1881,  on  a  note  dated  Jan.  4, 
1879,  for  $215.50,  with  interest  payable  annually  at  5J%,  when  no 
interest  has  been  paid.  * 

yrs.       moi.    dys.  $215.50 

1881      5    27 


1879      1      4 

2      4    23    0.143f 


12)30.996 
2.583 


$28.41  interest  for  2  yrs.  4  mos.  23  dys. 


teachers'  edition.  345 


yrs.    mos.    dys.  $215.50 

1      4     23 


4      23 


16  =  lift  yrs. 


0.051 
$11.85  annual  interest. 


$0.6518 
lift 


$1.17  interest  on  annual  interest. 
28.41 


$29.58  total  interest. 
215.50 


$245.08  Ans. 

48.  Find  the  amount  due  Jan.  16,  1881,  on  a  note  dated  Jan.  8, 
1879,  for  $3115.20,  v/ith  interest  payable  annually  at  5%,  when  no 
interest  has  been  paid. 


yrs.         mos.    dys. 

1881       1     16 
1879      1      8 

2      0      8    0.10^ 

$3115.20 
0.10^ 

$314,981  interest  for  2  yrs.  8  dys. 

yrg.    mos.    dys. 
10         8 

$3115.20 
0.05 

8 

$155.76  annual  interest. 

1      0     16  =  1^^  yrs. 

0.05 

$7,788 

h\ 

$8,134  interest  on  annual  interest. 

314.981 

$323.12  interest. 

3115.20 

$3438.32  Ans. 

49.  Find  the  amount  due  Jan.  18,  1881,  on  a  note  dated  Jan.  8, 
1877,  for  $2875,  at  6%:  (1)  simple  interest;  (2)  annual  interest; 
(3)  compound  interest. 


346 

ARITHMETIC. 

(1) 

yrs.        mo8.      dya. 

1881       1       18 
1877      1        8 

4      0      10 

yra. 

3 
2 
1 

(2) 

moB.      dys. 

0      10 
0      10 
0      10 

10 

(3) 

$2875 
0.06 

$172.50 
2875.00 

0.241f 

$2875 

6      1      10 

=  6Urs. 
Simple  interest 

=  $694.79. 
$2875 
0.06 

$3047.50 
0.06 

0.24  If 

$694.79 
2875.00 

$3569.79  Ans. 

$182.85 
3047.50 

$3230.35 
0.06 

$3633.04  Ans. 


$172.50  $193.82 


Q-Q^  3230.35 
$10.35  $3424.17 
H  0.06 

$63.25  $205.45 

694.79  3424.17 

2875.00 


$3629.62 
O.OOlf 

$6.05 
3629.62 

$3635.67  A71S. 


EXBRCISE  LXXIII. 

1.  Find  the  cost  of  $4000  stock,  at  109|. 

If  $1  stock  cost  $1.09J.  $4000  stock  will  cost 
4000  X  $  1.09  J  =  $4395.  Aiis. 

2.  Find  the  cost  of  $2600  stock,  at  98. 

If  $  1  stock  cost  $0.98,  $2500  stock  will  cost 
2500  X  $0.98  =  $2450.  Aiu. 


teachers'  edition.  347 

3.  Find  the  cost  of  1 3900  stock,  at  78|-. 

If  |1  stock  cost  |0.78i  13900  stock  will  cost 
3900  x$0.78|  =  $3046.88.  Ans. 

4.  Find  the  cost  of  1 4700  stock,  at  100^. 

If  |1  stock  cost  $1.00-1,  14700  stock  will  cost 
4700  X  1 1.00i  =  $  4723.50.  Ans. 

5.  Find  the  cost  of  $  1250  stock,  at  87f ,  brokerage  |-. 
If  f  be  the  brokerage,  87f  +  i  =  87f  is  total  cost. 
If  $1  stock  cost  |0.87f,  $1250  stock  will  cost 

1250  X  $0.87f  =  $  1096.88.  Ans. 

6.  How  much  bank  stock,  at  75^,  may  be  bought  for  $8729? 

If  $0.75^  buys  $1  stock,  $8729  will  buy  ^^  =  $11600.    Ans. 

7.  How  much  railroad  stock,  at  91^  may  be  bought  for  $4237^^? 
If  $0.91^  buy  $  1  stock,  $4237t\  will  buy  ^'^^^h^  =  $4650. 

8.  How  much  railroad  stock  may  be  bought  for  $6305,  at  121  J? 
If  $1.21|  buy  $1  stock,  $6305  will  buy  |^  =  $5200.  Ans. 

9.  How  much  railroad  stock  may  be  bought  for  $  5137.50,  at  102f  ? 
If  $  102f  buy  $  1  stock,  $  5137.50  will  buy  ^^^f^''^^  =  $  5000. 


10.    How  many   $100  railroad   shares,  at  68|,  may   be   bought 
for  $1650? 

If  1  share  cost  $68f ,  ^1^  =  24  shares  can  be  bought  for  $  1650. 


11.   What  must  be  the  price  of  stock,  in  cider  that  $9200  stock 
may  be  bought  for  $8970? 

If  $  9200  stock  cost  $  8970, 

then  $  1  stock  costs  $8970  -^  9200  =  0.97^ 
Hence,  the  stock  is  quoted  at  97|.  Ans. 


348  ARITHMETIC. 


12.  If  $1500  stock  be  bought  for  |1374.375,  what  is  the  price  of 
the  stock  ? 

If  $  1500  stock  cost  1 1374.375, 

then  $  1  stock  costs  $  1374.375  -;- J500  =  $0.91|. 
Hence,  the  stock  is  quoted  at  91f.  Ans. 

13.  What  income  will  be  derived  from  $29,700  4%  stock? 
0.04of$29,700  =  $1188.  Ans.  ^ 

14.  Find  the  income  from  $4500  6%  stock. 
0.06  of  $4500  =  $270.  Ans. 

15.  How  much  will  a  person  receive  from  $9400  railroad  stock  if 
a  dividend  of  3^  %  be  declared  ? 

0.035  of  $  9400  =  $  329.  Ans. 

16.  How   much   8%  stock   must  be  bought  to  give  an  income 
of  $2400? 

Since  $0.08  is  derived  from  $1  stock,  $2400  will  be  derived  from 
$2400 -^  0.08  =  $30,000.  Ans. 

17.  A  person  receives  $343  as  his  semi-annual  dividend  from  a 
7%  stock.     How  much  stock  does  he  hold  ? 

Since  $0.07  is  derived  from  $1  stock,  2  x  $ 343  =  $ 686  will  be 
derived  from  $686  -5-  0.07  =  $9800.  Ans. 

18.  Find  the  total  income  of  a  person  whose  property  consists  of 
$3000  6%  stocks  and  $8200  7%  stocks. 

0.06  of  $3000  =  $  180,  income  from  6%  stocks. 
0.07  of  $8200  =  $574,  income  from  7%  stocks. 
$180  +  $574  =  $754,  total  income.  Ans. 

19.  Find  the  rate  of  dividend  paid  by  some  stock,  when  a  holder 
of  $24,600  receives  $924.50. 

If  100  bo  taken  to  represent  $24,600,  the  number  required  to 

represent  $924.50  will  be  |^"V''^  oi  100  =  3J||. 
$  24600 

That  is,  mi%.  Am. 


teachers'  edition.  349 

20.  Find  the  rate  per  cent  at  which  $11,100  will  yield  a  semi- 
annual return  of  $499.50. 

If  100  be  taken  to  represent  $11,100,  the  number  required  to 

represent  2  X  $499.50  =  $999  will  be  r^^  of  100  =  9. 

$  11100 

That  is,  9%.  Ans. 

21.  If  $19,500  be  invested  in  4%  stock,  at  91,  what  income  will 
be  received  ? 

$91  is  the  cost  of  $100  stock. 

Hence,  $  1 9,500  =  cost  of  ^^^^  =  $  21,428.57  stock. 
'^     '  0.91 

And  0.04  of$21,428.57  =  $857.14.  Ans. 

22.  Find  the  income  on  $  7000  when  invested  in  4%  stock,  at  103^. 

$  103|  is  the  cost  of  $  100  stock. 

Hence,  $  7000  -  cost  of  $  7000  ^  1 .031  =  $  6779.66. 

And  0.04  of  $6779.66  =  $  271.19.  Ayis. 

23.  What  income  will  be  derived  from  $6800  if  it  be  invested  in 
7%  stocks,  at  130? 

$  130  is  the  cost  of  $  100  stock. 

Hence,  $6800  -  cost  of  $6800  -^  1.30  =  $5230.77. 

And  0.07  of  $5230.77  =  $366.15.  Ans. 

24.  A  person  invests  $7650  in  railroad  stock,  at  63|.     What  will 
he  receive  if  a  dividend  of  3|%  be  declared? 

$63|  is  the  cost  of  $100  stock. 

Hence,  $  7650  =  cost  of  $  7650  ^  0.63f  =  $  12,000. 

And  0.03^  of  $  12,000  =  $  390.  Ans. 

25.  If  3%  stocks  are  at  88|,  what  rate  per  cent  interest  will  a 
purchaser  receive  on  his  money  ? 

$  100  stock  costs  $  88^.     $  100  stock  pays  $  3. 

Hence,  the  $88^  invested  pays  $3. 

Therefore,  the  rate  of  interest  is  3  -r-  88^  =  0.03^f  or  3^f  %.  Aiis. 


350  ARITHMETIC. 


26.  If  an  8%  stock  is  at  150,  what  rate  per  cent  interest  will  a 
purchaser  receive  on  his  money  ? 

$  100  stock  costs  1 1 50.     $  100  stock  pays  1 8. 

Hence,  the  1 150  invested  pays  $8. 

Therefore,  the  rate  of  interest  is  8  -4- 150  =  0.05^  or  5|  %.  Ans. 

27.  If  a  10%  stock  is  at  175,  what  rate  per  cent  interest  will  an 
investor  receive  on  his  money  ? 

$  100  stock  costs  $  175.     $  100  stock  pays  1 10. 

Hence,  the  $  175  invested  pays  $  10. 

Therefore,  the  rate  of  interest  is  10  -r- 175  =  0.05^  or  of  %.  Ans. 

28.  If  a  41%  stock  is  at  85,  what  rate  per  cent  interest  will  a 
purchaser  receive  on  his  money? 

$  1 00  stock  costs  $  85.     $  100  stock  pays  1 4|. 

Hence,  the  |85  invested  pays  $4^. 

Therefore,  the  rate  of  interest  is  4^  -f-  85  =  0.05/^  =  5^.  %.  Ans. 

29.  If  a  7%  stock  is  at  114,  what  rate  per  cent  interest  will  a  pur- 
chaser receive  on  his  money? 

$  100  stock  costs  $  1 14.    1 100  stock  pays  $  7. 

Hence,  the  $  1 14  invested  pays  $7. 

Therefore,  the  rate  of  interest  is  7  -5- 114  -=  O.OG^*,-  or  ^z\%-  -^^«- 

30.  If  a  0%  stock  is  at  130,  what  rate  per  cent  interest  will  a  pur- 
chaser receive  on  liis  money  ? 

$100  stock  costs  $  130.     $  100  stock  pays  |6. 

Hence,  the  ^  130  invested  pays  $6. 

Therefore,  the  rate  of  interest  is  6  -^  130  =  OM/^^  or  ij%%-  Ans. 

31.  If  an  8%  stock  is  at  140,  what  rate  per  cent  interest  will  a 
purchaser  receive  on  his  money? 

$100  stock  costs  $140.     $100  stock  pays  $8. 

Hence,  the  $140  invested  pays  $8. 

Therefore,  the  rate  of  interest  is  8  +  140  =  0.05^  or  5^  %   Am, 


teachers'  edition.  351 


32.  How  much  money  must  be  invested  in  4%  stock,  at  92,  to 
receive  $245  income? 

$  100  stock  costs  1 92.    $  100  stock  pays  1 4. 
Hence,  the  $92  invested  pays  $4. 

Therefore,  the  rate  of  interest  is  4  h-  92  =  0.042^3  or  ^^^%. 
Since  $0.04^3  is  derived  from  $  1  stock,  $245  will  be  derived  from 
$  245  ^  0.04^3  =  $  5635.  .Ins. 

33.  Find  the  sum  required  for  an  investment  in  a  4%  stock,  at 
98^-,  to  produce  an  income  of  $200  a  year. 

$  100  stock  cost  $  98|.     $  100  stock  pays  $  4. 
Hence,  the  $98|  invested  pays  $4. 

Therefore,  the  rate  of  interest  is  4  h-  98|  =  0.04^^7  or  ^^if%. 
Since  %OM^^j  is  derived  from  $1  stock,  $200  will  be  derived 
from  $  200  -^  0.04x^7  =  $  4925.  Ans. 

34.  A  person  bought  some  bank  stock  at  107,  and  received  $384.25 
when  a  dividend  of  7|%  was  paid.     How  much  had  he  invested? 

Since  $0.07i  is  derived  from  $1  stock,  $384.25  will  be  derived 

from  $384.25  -  0.07^  =  $5300. 
$5300  is  the  cost  of  53  shares  at  100. 
Therefore,  the  cost  of  53  shares  at  107  will  be  53  x  $107  =  $5671. 

35.  What  must  be  the  price  of  a  5%  stock,  in  order  that  a  buyer 
may  receive  6%  on  his  money  ? 

If  $100  pay  $5  at  5%,  the  number  of  dollars  required  to  pay  $5 

at  6  %  will  be  |  of  $  100  =  $83|. 
That  is,  the  stock  is  quoted  at  83^.  Ans. 

36.  What  must  be  the  price  of  a  7%  stock,  in  order  that  a  buyer 
may  receive  6%  interest  on  his  money  ? 

If  $100  pay  $7  at  7%,  the  number  of  dollars  required  to  pay  $7 

at  6%  will  be  I  of  $  100  =  $  116f. 
That  is,  the  price  of  the  stock  will  be  116f.  Ans. 

37.  What  may  be  paid  for  an  8%  stock,  in  order  that  a  buyer 
may  receive  6%  interest  on  his  money?  for  a  9%  stock?  for  a  10% 
stock? 


352  ARITHMETIC. 


If  $100  pay  1 8  at  8%,  the  number  of  dollars  required  to  pay  f  8 

at  6%  will  be  f  of  $100  =  $  133^. 
That  is,  the  price  of  the  stock  will  be  133J.  (I)  Ans. 
If  $100  pay  $9  at  9%,  the  number  of  dollars  required  to  pay 

$9  at  6%  will  be  f  of  $  100  =  $  150. 
That  is,  the  price  of  the  stock  will  be  150.  (2)  Ans. 
If  $  100  pay  $10  at  10%,  the  number  of  dollars  required  to  pay 

$  10  at  6%  will  be  -^^  of  $  100  =  $  166f . 
That  is,  the  price  of  the  stock  will  be  166f .  (3)  Am. 

38.  A  person  invested  $2855  in  a  bank,  when  the  stock  was  at 
142|.  What  is  the  rate  per  cent  of  the  dividend  when  he  receives 
$150? 

The  number  of  shares  is  $  2855  -s-  $  142|  =  20. 

If  20  shares  pay  $  150,  1  share  will  pay  ^^  =  $  7^. 

That  is,  1\%.  Arts. 

39.  How  much  will  be  received  for  some  3%  stock,  from  which 
an  income  of  $250  has  been  derived,  if  sold  at  87^? 

Since  $0.03  is  derived  from  $1  stock,  $250  will  be  derived  from 

$250  ^  0.03  =  $8333.33  =  83|  shares. 
83}  shares  at  $87}  will  cost  83}  x  $87}  =  $7291.67.  Am. 

40.  If  a  5%  stock  pays  $340  income,  and  is  sold  out  for  $7990. 
at  what  price  is  it  sold  ? 

Since  $0.05  is  derived  from  $1  stock,  $340  will  be  derived  from 

$340  ^0.05  =  $6800. 
If  $6800  stock  be  sold  for  $7990,  then  $  1  is  sold  for 

$7990  ^6800  =  $1.17}. 
Hence,  the  stock  is  sold  at  117}.  Am. 

41.  On  what  per  cent  stock  must  an  investment  have  been  made 
from  which  $  185.50  was  derived  yearly,  and  which,  when  sold  out 
at  97,  brought  $5141? 

If  97  represent  $5141,  100  will  represent  ^  of  $5141  =  $5300. 
The  interest  on  $5300  for  1  year  is  $  185.50  ; 

on  $1  for  1  year  is  ?M:l^  =  $0,035. 
^  $5300 

Hence,  the  rate  is  3^%.  Am. 


teachers'  edition.  353 

42.  A  person  receives  4|%  interest  on  his  money  by  investing  it 
in  some  6%  stock.     At  what  price  did  he  buy  it? 

If  $100  pay  $6  at  6%,  the  number  of  dollars  required  to  pay 

$6  at  4i%  is  —  of  $  100  =  1 144. 
H 
Therefore,  the  price  of  the  stock  was  144.  Ans. 

43.  If  14800  3%  stocks  be  sold  at  88,  and  the  proceeds  be  invested 
in  5%  stocks  at  106f,  what  additional  income  will  be  obtained? 

0.03  of  1 4800  =  $  144,  income  from  the  3 %  stock. 

0.88  of  $4800  -14224,  amount  from  the  3%  stock. 

$1,056  is  paid  for  $1  worth  of  5%  stock. 

Hence,  $4224  is  paid  for  $4224  --  1.056  =  $4000  stock. 

0.05  of  $4000  =  $200,  income  from  5%  stock. 

$200  —  $  144  =  $56,  increase  in  income.  Ans. 

44.  If  $  7800  3 1  %  stocks  be  sold  at  60,  and  the  proceeds  be  invested 
in  5%  stocks  at  90,  find  the  alteration  in  income. 

0.03|  of  $7800  =  $273,  income  from  the  3^  %  stock. 
0.60  of  $7800  =  $4680,  amount  from  the  3^  %  stock. 
$0.90  is  paid  for  $  1  worth  of  5%  stock. 
Hence,  $4680  is  paid  for  $4680  --  0.90  =  $5200. 
0.05  of  $5200  =  $  260,  income  from  5 %  stock. 
$273  —  $260  =  $  13,  decrease  in  income.  Ans. 

45.  If  $10,000  3%  stocks  be  sold  at  88,  and  the  proceeds  be  in- 
vested in  3|%  stock  at  par,  find  the  alteration  in  income. 

0.03  of  $10,000  =  $300,  income  from  the  3%  stock. 
0.88  of  $  10,000  =  $8800,  amount  from  the  3%  stock, 
0.03|  of  $8800  =  $308,  income  from  the  3i%  stock. 
$308  -  $  300      =  $8,  increase  in  income.  Ans. 

46.  If  $10,000  8%  stocks  be  sold  at  150,  and  the  proceeds  be  in- 
vested in  6%  stocks  at  par,  find  the  alteration  in  income. 

0.08  of  $  10,000  =  $800,  income  from  the  8%  stock. 
1.50  of  $  10,000  =  $  15,000,  amount  from  the  8  %  stock. 
0.06  of  $  15,000  =  $900,  income  from  the  6%  stock. 
$900  -  $  800      =  $100,  increase  in  income.  Ans. 


354  ARITHMETIC. 


47.  If  $8000  10%  stocks  be  sold  at  170,  and  the  proceeds  be  in- 
vested in  5%  stocks  at  68,  find  the  alteration  in  income. 

0.10  of  $8000  -  $800,  income  from  the  10%  stock. 

1.70  of  $8000  =  $13,600,  amount  from  the  10%.«tock. 

$0.68  is  paid  for  $  1  worth  of  5%  stock.     Hence,  $  13,600  is  paid 

for  $  13,600  -^  0.68  =  $20,000  stock. 
0.05  of  $20,000  =  $  1000,  income  from  the  5%  stock. 
$  1000  -  $800  =  $200,  increase  in  income.  Ans. 

48.  If  $7000  8%  stocks  be  sold  at  150,  and  the  proceeds  be  in- 
vested in  6%  stocks,  at  105,  find  the  alteration  in  income. 

0.08  of  $  7000  =  $560,  income  from  the  8%  stock. 

1.50  of  $7000  =  $10,500,  amount  from  the  8%  stojk. 

$1.05  is  paid  for  $1  worth  of  6%  stock.     Hence,  $  10,500  is  paid 

for  $  10,500  -^  1.05  =  $  10,000  stock. 
0.06  of  $10,000  =  $600,  income  from  the  6%  stock. 
$600  —  $560  =  $40,  increase  in  income.  Am. 

49.  If  $1000  8%  stocks  be  sold  at  170,  and  the  proceeds  be  in- 
vested in  5%  stocks  at  par,  find  the  alteration  in  income. 

0.08  of  $1000  =  $80,  income  from  the  8%  stock. 
1.70  of  $  1000  =  $1700,  amount  from  tho  8%  stock. 
0.05  of  $1700  =  $85,  income  from  the  5%  stock. 
$85  —  $80  =  $5,  increase  in  income.  Ana. 

50.  If  $8000  5%  stocks  be  sold  at  90,  and  the  proceeds  be  invested 
in  3^%  stocks  at  60,  find  the  alteration  in  income. 

0.05  of  $8000  =  $400,  income  from  the  5%  stock. 

0.90  of  $8000  =  $  7200,  amount  from  the  5%  stock. 

$0.60  is  paid  for  $  1  worth  of  3  J  %  stock.     Hence,  $7200  is  paid 

for  $  7200  -5-  0.60  =  $  12,000  stock. 
0.03.1  of  $  12,000  =  $  420,  income  from  the  31  %  stock. 
$420  —  $400  =  $20,  increase  in  income.  Ans. 

51.  If  $10,000  3i%  stocks  be  sold  at  65.  and  the  proceeds  be  in- 
vested in  8  %  stocks  at  130,  find  the  alteration  in  income. 


teachers'  edition.  355 

0.03i  of  $  10,000  =  $  350,  income  from  the  3i  %  stock. 

0.65  of  1 10,000    =  $  6500,  amount  from  the  3i  %  stock. 

$1.30  is  paid  for  $  1  worth  of  8%  stock.     Hence,  $6500  is  paid 

for  $6500  ^  1.30  =  $5000  stock. 
0.08  of  $5000  =  $400,  income  from  the  8%  stock. 
$400  — $350  =$50,  increase  in  income.  Ans. 

52.    If  $8000  4^%  stocks  be  sold  at  70,  and  the  proceeds  be  in- 
vested in  10%  stocks  at  160,  find  the  alteration  in  income. 

0.04|  of  $8000  =  $360,  income  from  the  4i%  stock. 

0.70  of  $8000    =  $5600,  amount  from  the  4^  %  stock. 

$1.60  is  paid  for  $1  worth  of  10%  stock.     Hence,  $5600  is  paid 

for  $5600  --  1.60  =  $3500  stock. 
0.10  of  $3500  =  $350,  income  from  the  10%  stock. 
$  350  =  $  10,  decrease  in  income.  Ans. 


53.  If  $6000  6%  stocks  be  sold  at  90,  and  the  proceeds  be  invested 
in  10%  stocks  at  135,  find  the  alteration  in  income. 

0.06  of  $6000  =  $360,  income  from  the  6%  stock. 

0.90  of  $6000  =  $5400,  amount  from  the  6%  stock. 

$  1.35  is  paid  for  $  1  worth  of  10%  stock.     Hence,  $5400  is  paid 

for  $5400  ^  1.35  =  $4000  stock. 
0.10  o'f$ 4000  =  $400,  income  from  the  10%  stock. 
$400  —  $360  =  $40;  increase  in  income.  Ans. 

54.  Find  the  rate  of  interest  obtained  by  investing,  in  a  stock, 
at  124,  paying  6^  per  cent  per  annum. 

$100  stock  costs  $  124.     $  100  stock  pays  $6|. 

Hence,  the  $124  invested  pays  $6^. 

Therefore,  the  rate  of  interest  is  6^  -j-  124  =  O.OSyf^  or  5^1  j%. 

55.  What  is  the  price  of  stock  if  $  7000  stock  can  be  bought  for 
$5880? 

If  $  7000  stock  cost  $5880,  $  1  stock  costs  $5880  -=-  7000  =  $0.84. 
Hence,  the  stock  is  quoted  at  84.  Aiis. 

56.  Find  the  price  of  mining  shares  issued  at  $15  a  share  and 
sold  at  2^%  discount. 

$  15  -  0.02^  of  $  15  =  $  15  -  $  0.33f  =  $  14.66^.  Ans. 


856  ARITHMETIC. 


57.  How  much  3|%  stock  must  be  sold  at  81|,  in  order  to  buy 
$5000  4%  stock  at  94^  ;  brokerage,  }  in  each  transaction  ? 

If  ^  be  the  brokerage,  94^  +  i  =  94f ,  price  to  buyer. 
If  ^  be  the  brokerage,  81J  —  |  =  81  f,  price  to  seller. 
Therefore,  as  much  stock  must  be  sold  as 

— ^  of  $  5000  =  $  5787.46.  Aiis. 
81| 

58.  How  much  stock  must  be  sold  at  96|  to  raise  a  sufficient  sum 
for  discounting  a  note  lor  $1000,  due  49  days  hence,  and  discounted 
at  5.1%? 

The  discount  is  the  interest  on  $1000  for  52  days. 
Therefore,  the  discount  is  $  1000  X  0.00794  =  $7.94. 
And  the  proceeds  is  $  1000  -  $7.94  =  $992.06. 
If  $0.96875  buy  $1  stock,  $992.06  will  buy 
$992.06  -5-  0.96875  -  $  1024.06.  Ans. 

59.  A  broker  bought  $5000  stock  at  88^.  At  what  price  must  he 
sell  it  to  gain  $100? 

He  paid  $5000  X  0.885  =  $4425. 

To  make  $  100,  he  must  sell  for  $4425  +  $  100  =  $4525. 

Therefore,  he  must  sell  at  |^^  =  0.905  or  90^  Ans. 

60.  If  a  broker  buy  stock  at  85,  at  what*price  must  he  sell  it  to 
make  12]%  profit ;  brokerage,  ^  on  each  transaction. 

If  the  brokerage  be  |,  the  price  to  the  buyer  is  85j^. 
If  the  brokerage  be  |,  the  selling  price  must  be 

12|^  +  I  =  12|%  above  the  cost. 
Therefore,  the  selling  price  must  be 

1.1 2f  of  85.125  =  95f  nearly.  Ans. 

61.  "Which  is  the  more  profitable  stock  for  investment,  a  4  %  at 
85  or  a  3%  at  63 ?  a  3J%  stock  at  67J  or  a  4%  stock  at  81^? 

4%  at  85  =  4}f  %  at  100 ;  and  3%  at  63  =  4^f  %  at  100. 

4if  ^^  greater  than  4}^. 

Therefore,  3%  at  63  is  the  better  investment.  (1)  Atis. 

H%  at  67J  =  5^,5^%  at  100 ;  and  4%  at  8U  ^  4Hf 

5^^5  is  greater  than  4|f  f . 

Therefore,  3J%  at  67^  is  the  better  investment.  (2)  Ans. 


teachers'  edition.  357 

62.  Find  the  price  of  a  4^  %  stock  to  equal  a  3|%  stock  at  88|. 

1 100  stock  costs  $  88^    $  100  stock  pays  $  3|. 

Therefore,  the  $88^  invested  pays  |3i. 

Hence,  the  rate  of  interest  is  3i  -i-  88i  =  O.OSiff,  or  3|f  ?  %. 

If  $100  pay  $3}f-f  at  3iff  %,  the  number  of  dollars  required  to 

pay  mn  at  H%  wUl  be  -^  =  1.13^. 

That  is,  the  stock  will  cost  113^|.  Ans. 

63.  Find  the  price  of  a  5%  stock  to  equal  a  3%  stock  at  89|. 

1 100  stock  costs  $  89|.     $  100  stock  pays  1 3. 

Therefore,  the  $89^  invested  pays  $3. 

Hence,  the  rate  of  interest  is  3  -;-  89^  =  0.03Y^y3:j,  or  ^y7%%. 

If  1 100  pay  ^^i^f%  at  Sj%%%,  the  number  of  dollars  required  to 

pay  |3tV9-  at  5o/e  will  be  -|^  =  1.49i. 

That  is,  the  5%  stock  will  cost  149|.  Ans. 

64.  Find  the  price  of  a  3  ^  %  stock  to  equal  a  6  %  stock  at  par. 

If  $  100  pay  f  6  at  0%,  the  number  of  dollars  required  to  pay 

1 6  at  31%  will  be  —  =  0.58^. 
6 

That  is,  the  3|%  stock  will  cost  58|-.  Ans. 

65.  Find  the  profit  or  loss  in  buying  1 80,000  stock,  at  91|,  and 
selling  at  90  ;  brokerage,  |-  on  each  transaction. 

The  loss  on  $  1  is  $  0.91f  +  $  0.00^  -  ($ 0.90  -  $  O.OOi)  =  $  0.01|. 
The  loss  on  $80,000  is  0.01|  of  $80,000  =  $1500.  Ans. 

66.  "Which  is  the  better  investment,  a  5%  stock  at  137^,  or  a  3^% 
stock  at  91|?  What  rate  of  interest  would  be  received  from  each 
investment  ? 

5%  at  137i  =  3|f|%  at  100;  and  3i%  at  91i  =  3if  i%  at  100. 

2x8  3  '^^  greater  than  3f  f  |. 

Therefore,  3^%  stock  at  91|^  is  the  better  investment.  Ans. 


358  ARITHMETIC. 


67.  A  person  invests  ^  7370  in  the  purchase  of  a  stock  at  92.  What 
loss  will  he  sustain  if  he  sell  at  90,  brokerage  being  ^  in  each 
transaction  ? 

If  I  be  the  brokerage,  92  +  |  =  92|^,  price  to  the  buyer. 

Hence,  1 0.92^  buys  $  1  stock ;  1 7370  buys  ^  "^T  =  ^  ^^^• 

If  I  be  the  brokerage,  90  —  ^  =  89|,  price  to  the  seller. 

Hence,  1 1  will  bring  $0.89|,  and  $8000,  8000x  $0.89J  =  |7190. 

$7370 -$7190  =  $180  loss.  Am. 


68.  How  much  stock  rauft  be  sold  at  90|  so  that  when  the  pro- 
ceeds are  invested  in  a  mortgage,  at  6%,  $543.75  a  year  may  be 
received  ? 

Since  $0.06  is  derived  from  $1  stock,  $543.75  will  be  derived 
from  $543.75  -=-  0.06  =  $9062.50. 

If  $  1  costs  $0.90f,  $9062.50  will  cost  ^^^^^-^^  =  1 10,000.  Ans. 
^^  0.90|        ^ 

69.  A  person  invests  f  of  his  money  at  6%,  f  at  41%,  and  the  rest 
at  6%.     What  per  cent  will  he  receive  on  the  whole  amount? 

fx6    =lf 

4^%.  Am. 
Exercise  LXXIV. 

1.  Find  the  cost  of  a  draft  on  New  York  for  $1100,  at  \  of  1% 
premium. 

$  11 00  xl.00i  =  $  1102.75.  Am. 


2.   Find  the  cost  of  a  draft  on  New  Orleans  for  $  1350,  at  i  of  1% 
discount.  / 

$  1350  X  1.00}  =  $1346.62.  Am. 


teachers'  edition.  359 

3.    Find  the  cost  of  a  draft  for  $1600,  payable  30  days  after  sight, 
when  exchange  is  ^  of  1%  premium,  and  interest  6%. 

$1600.00 
0.0055  of  $  1600  =  8.80  discount  at  6%  for  33  days. 

$1591.20  proceeds  of  draft. 
0.0025  of  $  1600    =  4.00  premium  at  ^  of  1  %. 

$  1595.20  cost  of  draft.  Arts. 


4.  Find  the  cost  of  a  draft  for  $500,  payable  60  days  after  sight, 
when  exchange  is  ^  of  1%  discount,  and  interest  7%. 

$500,000 
0.01225  of  $500    =  6.125  discount  at  7%  for  63  days. 

$493,875  proceeds  of  draft. 
0.005      of  $500    =  2.500  discount  at  ^  of  1  %. 

$491.38  cost  of  draft.  Ans. 

5.  Find  the  cost  of  a  draft  for  $1200,  payable  in  90  days  after 
sight,  when  exchange  ig  ^  of  1%  premium,  and  interest  7%. 

$1200.00 
O.OlSOf  of  $  1200  =  21.70  discount  at  7  %  for  93  days. 

$1178.30  proceeds  of  draft. 
0.005      of  $  1200  =  6.00  premium  at  -^  of  1  %. 

$1184.30  cost  of  draft.  Ans. 


6.  Find  the  cost  of  a  draft  for  $950,  payable  in  30  days,  when 
exchange  is  at  par,  and  interest  4|%. 

$950.00 
0.004125  of  $950  =  3.92  discount  at  4|%  for  33  days. 

$946.08  cost  of  draft.  Ans. 

7.  Find  the  cost  of  a  draft  for  $725,  payable  in  60  days,  when 
exchange  is  at  ^  of  1%  discount,  and  interest  5%. 


360  ARITHMETIC. 


$725.00 
0.00875  of  $725    =  6.34  discount  at  5%  for  63  days. 

$  718.66  proceeds  of  draft. 
0.0025    of  $725   =  1.81  discount  at  ^  of  1  %. 


$716.85  cost  of  draft.  Ans. 


8.   Find  the  cost  of  a  draft  for  $810,  payable  in  90  days,  when 
exchange  is  at  ^  of  1%  premium,  and  interest  5|%. 

$810.00 
0.0142^3j  of  $810  =  11.51  discount  at  5i  %  for  93  days. 

$  798.49  proceeds  of  draft. 
0.0025      of  $810  =  2.03  premium  at  ^  of  1  %. 

$800.52  cost  of  draft.  Ans. 


9.   Find  the  face  of  a  draft,  payable  30  days  after  sight,  that  can 
be  bought  for  $274,  when  exchange  is  at  par,  and  interest  6%. 

$0.0055  =  Discount  on  $  1  for  33  days,  at  6%. 
$  1  -  $  0.0055  =  $  0.9945.  proceeds  of  $  1. 
Therefore,  $274  +  0.9945  -  $275.52,  face  of  draft.  Ans. 


10.  Find  the  face  of  a  draft,  payable  60  days  after  sight,  that  can 
be  bought  for  $1250,  when  exchange  is  at  |-  of  1%  premium,  and 
interest  7%. 

$0.01225  =  Discount  on  $1  for  63  days,  at  7%. 
$  1  -  0.01225  =  $0.98775  proceeds  of  $  1. 
0.0025    premium  on  $  1. 

$0.99025  cost  of  $1. 
Therefore,  $  1250  +  0.99025  =  $  1262.31,  face  of  draft.  Ans. 


11.  Find  the  face  of  a  draft,  payable  60  days  after  date,  that  can 
be  bought  for  $1125,  when  exchange  is  at  ^  of  1%  discount,  and 
interest  5J%. 


teachers'  edition.  361 

$0.009625  =  Discount  on  1 1  for  63  days,  at  o^  %. 
$1-10.009625  =  $0.990375  proceeds  of  $1. 
0.0025       discount  on  $  1. 

$0.987875  cost  of  $1. 
Therefore,  $  1125  ^  0.987875  -  $  1138.81,  face  of  draft.  Ans. 

12.  Find  the  face  of  a  draft,  payable  30  days  after  date,  that  can 
be  bought  for  $520,  when  exchange  is  at  \  of  1%  premium,  and 
interest  4  %. 

$  0.00361  =  Discount  on  $  1  for  33  days,  at  4%. 
$  1  -  $0.0036f  =  $0.99631  proceeds  of  $  1. 
0.005       discount  on  $1. 

$1.00131-  cost  of  $1. 
Therefore,  $520  -=-  1.0013i  =  $519.31,  face  of  draft.  Ans. 

13.  Find  the  face  of  a  draft,  payable  90  days  after  date,  that  can 
be  bought  for  $10,000,  when  exchange  is  at  par,  and  interest  4|%. 

$0.011625  =  Discount  on  $1  for  93  days,  at  4^  %. 
$1-0.011625  =  $0.988375,  proceeds  of  $1. 
Therefore,  $10,000  ^0.988375  =  $10,117.63,  face  of  draft.  Ans. 


Exercise  LXXV. 

1.  Find  the  cost  of  a  draft  on  London  for  £320  10s.  6d,  when 
sterling  exchange  is  quoted  at  4.83. 

£320  10  s.  6  (^.  =  £320.525. 
$  4.83  X  320.525  =  $  1548.14.  Ans. 

2.  Find  the  cost  of  a  thirty -day  draft  on  London  for  £150,  when 
thirty -day  bills  are  quoted  at  4.82,  and  the  broker's  commission  is  i 
of  1  %  of  cost  of  draft. 

$4.82x150  =  $723. 
1%  of  $723  =$0.90. 
$723  +  $0.90  =  $  723.90.  Ans. 


362  ARITHMETIC. 


3.  Find  the  cost  of  the  following  draft,  when  sixty-day  bills  are 
quoted  at  4.81,  and  the  broker's  commission  is  |  of  1  %  of  cost  of 
draft : 

£500.  New  York,  Feb.  17.  1881. 

Sixty  days  after  sight  of  this  First  of  Exchange  (Second  and  Third 
of  the  same  tenor  and  date  unpaid),  pay  to  the  order  of  James  Wilson 
five  hundred  pounds,  value  received,  and  charge  to  account  of 

To  James  Sage  &  Co.,     "I  Simon  Morton  &  Co. 

London.  J 

$4.81x500  =  12405. 
1%  of  1 2405  =  $3.01. 
$  2405  +  $  3.01  =  $  2408.01 .  Am. 

4.  Find  the  face  of  a  draft  on  Glasgow  that  can  be  bought  for 
$2000,  when  sterling  exchange  is  quoted  at  4.84. 

$2000 -=-$4.84  =  413y\V 

£413^2^  =  £413  4«.  5.5  d.  Am. 

6.  Find  the  face  of  a  draft  on  Dublin  that  can  be  bought  for 
$  135.24,  when  sterling  exchange  is  quoted  at  4.83. 

$135.24 -f- $4.83  =  28. 

That  is,  £28.  Am. 

6.  How  large  a  draft  on  London  can  be  bought  for  $4000,  when 
exchange  is  quoted  at  4.86f  ? 

$4000  +  $4.86f  =  821ff 

£821fJ  =  £821  18 «.  4.3d  Am. 

7.  Find  the  cost  of  a  draft  on  Paris  for  8000  fr.,  when  Paris  ex- 
change is  quoted  at  5.12|,  and  brokerage  is  |  of  1%. 

O.OOJ  of  8000  fr.  =  10  fr. 
8000  fr.  +  10  fr.  =  8010  fr. 
8000h-5.12|  =  15G3.31. 
Hence,  draft  of  8000  fr.  =  $  1563.31.  Am. 


teachers'  edition.  363 

8.  Find  the  cost  of  a  draft  on  Paris  for  10,000  fr.,  when  Paris 
exchange  is  quoted  at  5.14. 

10,000  ^  5.14  =  1945.53. 
Hence,  draft  of  10,000  fr.  =  $  1945.53.  Ans. 

9.  How  large  a  sixty-day  draft  on  Paris  can  be  bought  for  $1500, 
when  sixty-day  bills  are  quoted  at  5.11|? 

1500x5.111=7672.5. 
That  is,  7672.5  fr.  Ans. 

10.  How  large  a  sight  draft  on  Paris  can  be  bought  for  $  2840, 
when  Paris  exchange  is  quoted  at  5.13|  ? 

2840x5.131=14,583.4. 
That  is,  14,583.4  fr.  Ans. 

11.  Find  the  cost  of  a  draft  on  Hamburg  for  2876  marks,  when 
German  exchange  is  quoted  at  0.95|. 

2_8^  of  10.951  _  1686.65.  Ans. 

12.  Find  the  cost  of  a  draft  on  Munich  for  12,000  marks,  when 
German  exchange  is  quoted  at  0.94f . 

i_2 000  of  |o.94f  =  $ 2838.75.  ^ns. 

13.  How  large  a  draft  on  Frankfort  can  be  bought  for  |1200, 
when  German  exchange  is  quoted  at  0.95|  ? 

0.95^  -^  4  =  0.23|. 
1200  ^  0.23|  =  5026.17  marks.  Am. 

Exercise  LXXVI. 


1.   If  a  dozen  eggs  weigh  692k, 
what  is  the  mean  weight  of  an 

egg? 

12)692g 

57f?.  Ans. 


2.  Seven  boys  weigh  respec- 
tively 119.7  lbs.,  105  lbs.,  178.3 
lbs.,  165.3  lbs.,  142.8  lbs.,  109 
lbs.,  154.2  lbs.  What  is  their 
average  weight? 


304 


ARITHMETIC. 


119.7 
105.0 
178.3 
165.3 
142.8 
109.0 
154.2 

7)974.3  lbs. 
139f^lb8.  Ans. 

3.  A  merchant  mixes  1  lb.  of 
coffee  worth  27  cents,  1  lb.  worth 
35  cents,  and  1  lb.  worth  40 
cents.  What  are  the  3  lbs.  to- 
gether worth  ?  How  much  a 
pound  is  the  mixture  worth  ? 

$0.27 
0.35 
0.40 

3)1.02 
10.34.  Ans. 

4.  A  merchant  mixes  2  lbs.  of 
coffee  worth  27  cents  a  pound,  3 
lbs.  worth  35  cents  a  pound,  and 
1  lb.  worth  40  cents.  What  is  a 
pound  of  the  mixture  worth  ? 

2  X  $0.27  =  $0.54 
3x  0.35=-^  1.05 
Ix    0.40=    0.40 


6)$  1.99 
$0.3aj.  Am. 

6.  What  is  the  value  per  pound 
of  a  mixture  of  coffee  containing 
7  lbs.  worth  26  cents  a  pound,  4 
lbs.  worth  31  cents  a  pound,  and 
10  lbs.  worth  34  cents  a  pound  ? 


7  X  $0.26  =  $1.82 

4x    0.31=    1.24 

10  X    0.34=    3.40 

21)$6.46 

$0.30|f.  Ans. 

6.  What  is  the  cost  of  a  gallon 
of  the  mixture  in  which  7  gals, 
cost  67  cents  a  gallon,  5  gals,  cost 
48  cents  a  gallon,  and  water, 
without  cost,  was  added  until 
there  were  15  gals,  of  the  mix- 
ture? 

7  X  $0.67  =  $4.69 
5x    0.48=    2.40 


15)  $7.09 
$0.47. 


Ans. 


7.  If  7'  of  water  be  poured 
into  a  vessel  containing  3*  of 
sulphuric  acid,  specific  gravity 
1.840,  and  the  mixture  shrink  to 
9.972^  what  is  the  specific  gravity 
of  the  mixture  ? 


7x1' 


.00> 


3  X  1.84  =  5.52 


9.972')  12.52' 

1.256.  Am. 

8.  If  4'  of  water  and  1'  of  sul- 
phuric acid,  specific  gravity, 
1.842,  when  mixed  shrink  \  of 
1%,  what  is  the  specific  gravity 
of  the  mixture  ? 

4x1'  =  4.000> 
1x1.842=1.842 

4.9875')  5.842» 
1.171* 

4  +  1  =  5. 
5-0.01^  =  4.9875.  An$. 


teachers'  edition.  365 

9.  In  what  proportions  must  tin  of  specific  gravity  7.29  and  lead 
of  specific  gravity  11.35  be  mixed  to  make  a  solder  of  specific  gravity 
10.21,  if  no  allowance  be  made  for  expansion  or  condensation  ?  (Give 
the  proportions  in  bulk.) 

The  specific  gravity  of  tin  lacks  2.92  of  the  required  specific 
gravity  ;  and  the  specific  gravity  of  lead  is  1.14  above  the 
required  specific  gravity. 

Therefore,  the  tin  is  to  the  lead  in  the  inverse  ratio  of  292  to 
114.     That  is. 

Tin  :  lead  =  114  :  292  =  57  :  146.  Ans. 

10.  In  what  proportions  must  oils  worth  $1.25  and  80  cents  a 
gallon  be  mixed  to  make  a  mixture  worth  $1  a  gallon?  (Test  the 
answer.) 

The  cost  of  the  better  oil  is  $0.25  above  the  required  cost;  and 
the  cost  of  the  worse  oil  lacks  $  0.20  of  the  required  cost. 

Therefore,  the  better  is  to  the  worse  in  the  inverse  ratio  of  $0.25 
to  $0.20.     That  is, 

The  better  :  the  worse  =  20  :  25  =  4  :  5.  Ans. 

11.  In  what  proportion  may  oils  worth  $1.20,  80  cents,  and  60 
cents  a  gallon  be  mixed  so  that  the  mixture  shall  be  worth  70  cents 
a  gallon  ? 

When  the  80-cent  oil  alone  is  taken,  in  what  ratio  to  the  60-cent 
must  it  be  used?  When  the  $1.20  oil  alone  is  taken,  in  what  ratio 
to  the  60-cent  oil  must  it  be  used?  When  the  $1.20  and  80-cent  oils 
are  mixed  gallon  for  gallon,  how  much  60-cent  oil  must  be  added  ? 
When  1  gal.  of  the  $1.20  oil  and  3  of  the  80-cent  oil  are  taken,  how 
much  60-cent  oil  must  be  added  ?  If  three-fourths  of  the  mixture 
consist  of  the  60-cent  oil,  what  per  cent  of  each  of  the  other  two 
must  be  taken  ? 

(1) 

The  cost  of  the  $1.20  is  $0.50  above  the  required  cost;  the  cost 
of  the  $0.80  oil  is  $0.10  above  the  required  cost;  and  the  cost 
of  the  $0.60  oil  lacks  $0.10  of  the  required  cost. 

Therefore,  the  $1.20  oil  is  to  the  $0.80  oil  is  to  the  $0.60  oil  in 
the  inverse  ratio  of  50  to  10  to  10.     That  is. 

The  $  1.20  oil :  the  $0.80  oil .  the  $0.60  oil  =  10 :  10 :  50  =  1 : 1  :^  L 


366  AEITHMETIC. 


(2) 
The  cost  of  the  $0.80  oil  is  $0.10  above  the  required  cost;  the 

cost  of  the  $0.60  oil  lacks  $0.10  of  the  required  cost. 
Therefore,  the  $0.80  oil  is  to  the  $0.60  oil  in  the  inverse  ratio  of 

10  to  10.     That  is. 
The  $0.80  oil :  the  $0.60  oil  =  10  :  10  =  1 :  1.  Am. 

(3) 
The  cost  of  the  $1.20  oil  is  $0.50  above  the  required  cost;  and 

the  cost  of  the  $0.60  oil  lacks  $0.10  of  the  required  cost. 
Therefore,  the  $1.20  oil  is  to  the  $0.60  oil  in  the  inverse  ratio 

of  50  to  10.     That  is, 
The  $  1.20  oil :  the  $0.60  oil  =  10  :  50  =  1 :  5.  Ans. 

(4) 
When  1  gal.  of  the  $1.20  oil  is  mixed  with  1  gal.  of  the  $0.80 
oil,  the  mixture  costs  $l'20  +  $0.80  _  ^j  qq  p^^  ^^-^^^^ 

A 

The  cost  of  the  $1.00  oil  is  $0.30  above  the  required  cost;  the 
cost  of  the  $0.60  oil  lacks  $0.10  of  the  required  cost. 

Therefore,  the  $1.00  oil  is  to  the  $0.60  oil  in  the  inverse  ratio 
of  30  :  10.     That  is, 

The  $  1.00  oil  is  to  the  $0.60  oil  =  10  :  30  =  1 :  3. 

1:3:  :  2  gals. :  what? 

1 :  3  :  :  2  gals. :  6  gals.  Ans. 

(5) 
When  1  gal.  of  the  $1.20  oil  is  mixed  with  3  gals,  of  the  $0.80 
oil,  the  mixture  costs  1^-20  +  3  x  $0.80  _  ^q^  ^^^  ^^ 

The  cost  of  the  $0.90  oil  is  $0.20  above  the  required  cost;  the 
cost  of  the  $0.60  oil  lacks  $0.10  of  the  required  cost. 

Therefore,  the  $0.90  oil  is  to  the  $0.60  oil  in  the  inverse  ratio 
of  20 :  10.     That  is. 


The  $0.90  oil 
1  :  2 :  :  4  gals. 
1 :  2  : :  4  eals. 


the  $0.60  oil  =  10:  20=  1:2. 

what? 

8  gals.  Ans. 

(6) 

Each  of  the  others  will  be     ""  ^  =  J  of  the  whole,  or  12J  %.  Ans. 


TEACHERS     EDITION. 


367 


12.  A  solder  composed  of  tin  and  lead,  specific  gravities  7.29  and 
11.35,  has  a  specific  gravity  of  10.44.  What  is  the  weight  of  each 
metal  in  a  kilogram  of  solder  ? 

The  specific  gravity  of  the  tin  lacks  3.15  of  the  required  specific 

gravity  ;  and  the  specific  gravity  of  the  lead  is  0.91  above  the 

required  specific  gravity. 
Therefore,  the  tin  is  to  the  lead  in  the  inverse  ratio  of  3.15  to 

0.91.     That  is, 
The  tin  :  the  lead  =  91  :  315.        91  +  315  =  406. 

5%V  of  10008  =  224/^«  tin.     fif  of  1000«  =  775|f«  lead.  Ans. 


13.  Find  the  equated  time  for 
the  payment  of  |250  due  in  3 
mos.,  $400  due  in  6  mos.,  $700 
due  in  8  mos. 

$250x0  = 
$400x3  =  11200 
$700x5  =  $3500 


$1350 


)$4700 
3if 


3|f  mos.  =  3  mos.  14  dys. 
3  mos.  14  dys.  after  3  mos. 
mos.  14  dys. 


14.  Find  the  equated  time  for 
the  payment  of  $300  due  in  30 
dys.,  $500  due  in  60  dys.,  and 
$  200  due  in  90  dys. 

$300x00  = 
$500x30  =  $15,000 
$200x60  =  $12,000 


$1000 


.27,000 
27 


Hence,  the  equated  time  is  27 
dys.  after  30  dys.  =  57  dys. 


15.  Find  the  equated  time  for 
the  payment  of  $325  due  now, 
$200  due  in  30  dys.,  $460  due  in 
60  dys.,  and  $  150  due  in  90  dys. 

$325x00  = 
$200x30  =  $  6,000 
$460x60  =  $27,600 
$  150  X  90  =  $  13,500 


$1135 


)$47,100 
41 


Hence,  the  equated  time  is  41 
dys. 

16.  Find  the  equated  time  for 
the  payment  of  $240  due  May 
10,  $420  due  July  2,  $310  due 
Sept.  14,  and  $600  due  Oct.  1. 

$240  X    00  = 
$420x    53  =  $22,260 
$310x127  =  $39,370 
$600x144  =  $86,400 


$1570 


)$  148.030 
94 


Hence,  the  equated  time  is  94 
dys.  after  May  10  =  Aug.  12. 


368 


ARITHMETIC. 


17.  Find  the  equated  time  for 
the  payment  of  |275  due  June 
21,  $175  due  July  16,  $200  due 
Aug.  6,  and  $150  due  Sept.  3. 

$275x00  = 
$175x25  =  $  4,375 
$200x46  =  $  9,200 
$150x74  =  $11,100 

$800  )$  24.675 

31 
Hence,  the  equated  time  is  31 
dys.  after  June  21  =  July  22, 


18.  Find  the  equated  time  for 
the  payment  of  $  112.30  due  July 
6,  $115.25  due  July  30,  $282.15 
due  Sept.  4,  and  $102.36  due 
Oct.  1. 

$112.30x00- 
$115.25x24  =  $  2,766.00 
$232.15x60  =  $13,929.00 
$102.36x87  =  $   8,905.32 


$562.06 


)$  25.600,32 
46 


Hence,  the  equated  time  is  46 
dys.  after  July  6  =  Aug.  21. 


19.  A  owed  B  $2000  payable  in  4  mos.,  but  at  the  end  of  1  mo. 
he  paid  him  $500,  at  the  end  of  2  mos.  $500,  and  at  the  end  of  3 
mos.  $500.     In  how  many  months  is  the  balance  due  ? 

$500x3  =  $1500 
$500x2  =  $1000 
$500x1=$  500 

$1500  $3000 

Therefore,  he  is  entitled  to  keep  the  balance  ($500)  -3^^  mos. 
=  6  mos.  after  its  maturity. 

20.  A  merchant  bought,  Feb.  11,  1881,  a  bill  of  goods  amounting 
to  $1700,  on  4  months'  credit;  but  he  paid  March  22,  $400,  April 
20.  $220,  May  10,  $300.    When  is  the  balance  due  ? 


1881 


dyi. 

11 


1881 


6 


11  =  June  11.  1881. 


$400x81  =  $32,400 

$220x52  =  $11,440 

$300  X  32  =  $  9,600 

$920  $53,440 

Therefore,  he  is  entitled  to  keep  the  balance  ($780)  *?f^*  dys 
69  dys.  after  its  maturity,  June  11,  1881,  =  Aug.  19,  1881. 


TEACHERS     EDITION. 


Find  the  equated  time  of  maturity  of  each  of  the  following  bills, 
and  the  amount  due  at  settlement,  including  interest  at  6%: 

21.  James  Pkice,  to  John  Bates,  Dr. 
1881. 

Apr.    5.     To  mdse.  on  4  mos.  credit     ...    $  120.50 
Apr.  15.  "         "   3    "        "         ...         87.33 

May    7.  "         "  3    "        "  ...       218.17 

May  21.  "         "  4    "        "         ...       317.00 

1743.00 
Paid  Oct.  18,  1881. 

$  87.33  X  00  =  From  Aug.  23  to  Oct. 

$120.50x21  =  $   2,530.50  18  is  56  dys. 

$218.17x23  =  $  5,017.91  56  dys.  =  O.OOQi 

$317.00x68  =  $21,556.00  $743 

$743.00  )$  29,104.41  ^'^^^^ 

39  $6.93 

Hence  the  equated  time  is  39  dys.  after  '__ 

Julyl5,  1881  =  Aug.  23,1881.  $749.93 

22.  Hall  &  Co.  bought  of  Boles  &  Co. 
1881. 

Feb.  11.     To  mdse.  on  30  dys $250.00 

Apr.  20.  "         "     2  mos 500.00 

May  31.  "  "     3  mos 150.00 

July    6.  "         "  60  dys 1000.00 

Paid  Nov.  10,  1881. 

30  dys.  after  Feb.  11  =  Mar.  13.  Hence,    the     equated 

2  mos.  after  Apr.  20  =  June  20.  time  is  132dys.  after  Mar. 

3  mos.  after  May  31  =  Aug.  31.  13, 1881  =  July  23, 1881. 
60  dys.  after  July    6  =  Sept.  4.  From  July  23  to  Nov. 

$  250  x    00=  10  is  110  dys. 

$  500  X    99  =  $  49,500  110  dys.  =  0.018i 
$   150x171  =  $  25,650  _$1900 

$1000x175  =  $175,000  $34.83 

$  1900  )$  250,150  i^QQQQ 

132  $1934.83 


370 


ARITHMETIC. 


23.   Find  the  equated  time  of  maturity  of  each  side  of  the  following 
account : 


Adams  &  Co.  in  account  vdth  Bacon  &  Co. 


Db. 


Ce. 


1881. 

Jan.  3. 

To  mdse.  90  dys. 

$250 

Apr.  11. 

By  cash, 

$200 

Mar.  7. 

"        60   " 

150 

Apr.  30. 

" 

100 

May  3. 

60   " 

325 

May  30. 

(t 

125 

June  7. 

"       30  " 

175 

July    2. 

" 

400 

90  dys.  after  Jan.  3  =  April  3. 
60  dys.  after  Mar.  7  =  May  6. 
60  dys.  after  May  3  =  July  2. 
30  dys.  after  June  7  =  July  7. 


De. 


$250x00  = 
$150x33  =  $  4,950 
$325x90  =  $29,250 
$175x95  =  $16,625 

$900  )$  50,825 

56 
Hence,  the  equated  time  is  56 
days  after  April  3  --  May  29. 


Ce 

$200  X 

$100  X 

$125  X 
$400  X 

00  = 
19^ 
49  = 
82  = 

=  $  1,900 
=  $  6,125 
=  $32,800 

$825 

)$40,825 
49 

Hence,  the  equated  time  is  49 
days  after  April  11  =  May  30. 


Find  the  time  for  paying  the  balance  in  the  following  equated  bills 


Average  due.  Dr. 

24.  May  17,  1881  .  .  .    $950 

25.  Apr.  12,  1881  .  .  .    $950 

26.  May  30,  1881  .  .  .  $1000 

27.  July  6,  1881  .  .  .    $500 


Average  due. 
Apr.  12,  1881    . 


Ce. 

$1000 


May  17.  1881  ....  $1000 
June  23,  1881  ....  $920 
Apr.  14,  1881  ....   $480 


teachers'  edition.  371 

24. 

Differences  in  equated  time  =  35  dys. 

Balance  of  account  =  1 1000  -  $  950  =  1 50. 

If  the  account  were  settled  at  the  later  date,  May  17,  the  $1000 
on  the  Cr.  side  would  have  been  on  interest  35  dys.,  and  this 
is  equivalent  to  having  the  balance,  1 50,  on  interest  -f ^-  of 
35  dys.  =  700  dys.  Hence,  the  balance  should  begin  to  draw 
interest  700  dys.  before  May  17,  1881 ;  that  is,  June  17,  1879. 

25. 

The  difference  in  equated  time  =  35  dys. 

Balance  of  account  =  1 1000  -  $  950  =  1 50. 

If  the  account  were  settled  at  the  later  date.  May  17,  the  $950 
would  have  been  on  interest  35  dys.,  which  is  equivalent  to 
having  the  balance,  1 50,  on  interest  ^-^^^-  of  35  dys.  =  665  dys. 

Hence,  to  increase  the  Cr.  side  by  an  equal  amount  of  interest, 
the  balance  should  remain  unpaid  665  dys.  ;  that  is,  the  bal- 
ance is  due  March  13,  1883. 

26. 

The  difference  in  equated  time  ==  24  dys. 

Balance  of  account  =  1 1000  -  |920  =  $80. 

If  the  account  were  settled  at  the  later  date,  June  23,  the  $  1000 
on  the  Dr.  side  would  have  been  on  interest  24  dys.,  and  this  is 
equivalent  to  having  the  balance,  $80,  on  interest  ^f^^  of  24 
dys.  =  300  dys. 

Hence,  the  balance  should  begin  to  draw  interest  300  dys.  be- 
fore June  23,  1881 ;  that  is,  Aug.  27,  1880. 

27. 

The  difference  in  equated  time  is  83  dys. 

Balance  of  account  =  $  20. 

If  the  account  were  settled  at  the  later  date,  July  6,  the  $480 
would  have  been  on  interest  83  dys.,  which  is  equivalent  to 
having  the  balance,  $20,  on  interest  -W-  of  83  dys.  =  1992  dys. 

Hence,  to  increase  the  Dr.  side  by  an  equal  amount  of  interest, 
the  balance  should  remairi  unpaid  1992  dys. ;  that  is,  the  bal- 
ance is  due  Dec.  19,  1886. 


372 


ARITHMETIC. 


Find  (by  either  method)  the  cash  balance  in  the  following  bills, 

reckoning  interest  at  6%: 

Qft 

1881.                         Dr. 

Int. 

1881.                         Cr. 

Int. 

Apr.    5.   To  mdse.     $250 

$3.17 

Apr.  20.   Bycafih,     $200 

$2.03 

"    27.         "             610 

5.49 

"    30.        "              500 

4.25 

June    1.          "              200 

0.63 

June   4.        "              400 

1.07 

"    20.    Tobal.acc.      40 

"    20.    Bybal.  int. 

1.94 

$1100 

$9.29 

$1100 

$9.29 

Hence,  cash  balance  =  $40  -  $1.94  =  $38.06.  Ans. 

29. 

1881.                        Dr. 

Int. 

1881.                         Cr. 

Int. 

Apr.  15.   To  mdse.  $250.00 

$7.42 

Apr.  26.  By  cash,  $150.00 

$4.18 

May  25.         "            98.50 

2.27 

May  17.        "          150.00 

3.65 

June  7.         "          300.00 

6.25 

July    7.        •'          200.00 
Oct.  10.  Bybal. ace.  148.50 

3.17 

•'     10.       "     int. 

4.94 

$648.50 

$15.94 

$648.50 

$15.94 

Hence,  cash  balance  =  $148.50  +  $4.94  =  $153.44.  Ans. 

30. 

1881.                       Dr. 

Int. 

1881.                        Cr. 

Int. 

Feb.  2.    To  mdse.     $100 

$3.02 

Feb.  25.    By  cash,     $100 

$2.63 

Apr.  7.          "              200 

3.90 

Mar.  22.         "              150 

3.33 

June  2.          ••                95 

0.97 

June  20.         "              200 

1.43 

"     9.                          150 

1.35 

Aug.    2.   Bybal.  ace.   95 

"     2.        "        int. 

1.85 

$545 

$9.24 

$545 

$9.24 

Hence,  cash  balance  =  $95  +  $1.85  =  $96.85.  Ans. 

31. 

1881.                       Dr. 

Int. 

1881.                        Cr. 

Int. 

Apr.   5.  To  mdse.      $250 

$6.21 

Apr.  20.   By  cash,     $200 

$4.47 

•'   27.         "              670 

14.18 

"   30.        ••              500 

10.33 

June  4.         "              200 

2.97 

June  1.        ••               400 
Sept.  1.   By  bal.  ace.    20 

6.13 

'•     1.       ••         int. 

$1120 

2.43 

$1120 

$23.36 

$23.36 

Hence,  cash  balance  = 

.$20  + 

J2.43  =  $22.43.  Am. 

TEACHERS     EDITION. 


373 


1881.  Dk. 

Mar.  10.  To  mdse.  |580 

Apr.  20.        "  200 

May    5.        "  150 

"    17.        "  325 


32. 

Int. 

$8.31 
1.50 
0.75 
0.98 


1881.  Ce. 

Mar.  15.  By  cash,    $500 

Apr.  15.  "              300 

"    25.  "              120 

May  20.  "              225 

June   4.  By  bal.  ace.  110 

"     4.  "       int. 


Int. 

$6.75 
2.50 
0.80 
0.56 

0.93 


$1255  $11.54  $1255   $11.54 

Hence,  cash  balance  =  $110  +  $0.93  =  $110.93.  Ans. 


Exercise  LXXVII. 


1.   Find   the    square  root  of 
2916. 

4.   Find   the    square   root  of 
20,164. 

29^6(54 
25 

104)416 
416 

2^01^64(142 
1 

24)101 
96 

282)564 

2.   Find   the    square  root  of 

7921. 

79^21(89 

64 
169) 1521 

564 

5.   Find   the    square   root  of 
3,345,241. 

1521 

3^34^52^41 (1829 
1 

3.   Find   the    square  root  of 
494,209. 

28)234 
224 

49^42^09(703 
49 

362)  1052 
724 

1403)4209 
4209 

3649)32841 
32841 

374 


ARITHMETIC. 


6.    Find    the  square    root  of 
125,457.64. 


8.   Find    the    square  root  of 
21,609. 

2^6^09(147 
1 

24)116 
96 

287)2009 
2009 


9.   Find    the    square   root  of 
53.7289. 


12^54^57.64(354.2 
9 

65)354 
325 

704)2957 
2816 

53.72^89(7.33 
49 

143)472 
429 

1463)4389 
4389 

7082) 14164 
14164 

7.   Find    the  square    root    of 
47,320,641. 

47^32^06^41(6879 
36 

128)1132 
1024 

10.   Find   the   square  root  of 
883.2784. 

8^83.27^84(29.72 
4 

49)483 
441 

587)4227 
4109 

1367)10806 
9569 

5942)11884 
11884 

13749) 123741 
123741 

11.   Find   the  square   root  of 
1.97262025. 

1.97'26'20^25(1.4045 
1 

24)97 
96 


2804)12620 
11216 


28085) 140425 
140425 


TEACHERS     EDITION. 


375 


12.    Find   the  square   root  of 
0.0002090916. 

0:00^02^09''09a6  (0.01446 
1 
24) 109 
96 
284)1309 
1136 
2886)17316 
17316 


13.   Find  the  square  root  of  2. 
2.00^00^00(1.414213 

24)100 
96 
281) 400 
281 
2824) 11900 
11296 


2828)6040 
5656 


3840 
2828 
10120 

8484 


14.    Find  the  square  root  of  5 
5.00^00^00(2.236067 


42)100 
84 
44:5)  1600 
1329 
4466)27100 
26796 
4472)"30400 
26832 


35680 
31304 


15.   Find  the  square  root  of  0.3. 

0.30^00^00^00(0.547722 
25 
104) 500 
416 


1087)8400 
7609 


10947) 79100 

76229 


10954) 24710 
21908 


28020 
21908 


16.   Find  the  square  root  of  3^. 

3.25^00^00(1.802775 
1 
28) 225 
224 


3602) 10000 
7204 
3604)27960 

25228 


27320 
25228 
20920 
18020 


17.   Find  the  square  root  of  8|. 

8.83^33^33(2.972092 

4_ 
49)483 

441 
587)4233 
4109 


6942)12433 
118S4 
54933 
53496 


14373 
11888 


376 


ARITHMETIC. 


18.    Find  the  square  root  of  0.9, 

0.90^00^00^00(0.948683 
81 

184) 900 
736 


1888)16400 
15104 


18966)129600 
113796 


18972) 158040 
151776 


62640 
56916 


19.   Find  in  yards  the  side  of 
a  square  field  containing  20  acres. 

20  A.  =  96,800  sq.  yds. 

9^68^00.00^00  (311.12 

9 

61)68 
61 
621)700 
621 


6221) 7900 
6221 


62222) 167900 
124444 


20.  Find  the  side  of  a  square 
the  area  of  which  is  150  sq.  ft. 
9  sq.  in. 

150  sq.  ft.  9  sq.  in.  =  21,609  sq.  in. 

2^6^09(147 

24)116 
96 

287) 2009 
2009 

147  in.  =  12  ft.  3  in. 


21.  Find  the  side  of  a  square 
the  area  of  which  is  8  sq.  yds.  7 
sq.  ft.  73  sq.  in. 

8  sq.  yds.  7  sq.  ft.  73  sq.  in. 

=  11,449  sq.  in. 

1'14'49(107 

1 


207) 1449 
1449 

107  in.  =  8  ft.  11  in. 


22.    Find  to  six  places  of  deci- 
mals  the  square  roots  of  | ;  f ; 

^;  I;  f;  f;  f;  I 
(1) 

V|  =  I  =  0.666667. 

(2) 
f  =  0.5. 

0.55^55'55'55^55^55  (0.745355 
49 

144)655 
576 


1485) 7955 
7425 


14903)53055 
44709 


83465 
74515 

89505 
74515 


TEACHERS     EDITION. 


377 


(3) 
1  =  0.5. 

0.50^00^00^00(0.707106 

49 

1407) 10000 
9849 


14141)15100 
14141 
14142)95900 

84852 


(4) 


0.60^00^00^00(0.774596 
49 
147)1100 
1029 


1544) 7100 
6176 


15485)92400 
77425 


15490)149750 
139410 


103400 
92940 


(5) 
f  =  0.714285714285. 

0.71^42^85^7F42'85  (0.845154 
64 
164) 742 
656 


1685)8685 
8425 


16901) 26071 
16901 


169025)917042 
845125 


1690304)7191785 
6761216 


(6) 
f  =  0.75. 
0.75^00^00^00(0.866025 

64 
6)1100 
996 


1726) 10400 
10356 


17320)44000 
34640 


93600 
86600 


(7) 
f  =  0.66666666. 

0.66^66^66^66(0.816496 
64 
161)266 
161 
1626) 10566 
9756 


16324)81066 
65296 


16328) 157706 
146952 


107546 
97968 


(8) 
f  =  0.833333. 
0.83^33^33(0.91287: 
81 
181)233 
181 


1822)5233 
3644 


1824) 15893 
14592 


13013 
12768 
2453 
1824 


378  ARITHMETIC. 


23.   Fin<l,  by  factoring,  the  square  roots  of  2025;  17.64;  2.0164; 
533.61:  204.49. 


2025  =  52  X  92. 
/2025  =  5  X  9  =  45.  (1)  Ans. 

17.64  =  62  X  0.72. 


.-.   VlfM  =  6  X  0.7  =  4.2.  (2)  Ans. 

2.0164  =  22  X  0.7P. 
.-.  V2.0164  =  2  X  0.71  =  1.42.  (3)  Ans. 

533.61  =  32  X  0.72  X  IP. 
/.  V5K6I  =  3  X  0.7  X  11  =  23.1.  (4)  Ans. 

204.49  =  IP  X  1.32. 


.-.  \/20l49  =  11  X  1.3  =  14.3.  (5)  Am.     - 

24.  A  ladder  13  ft.  long  standing  on  level  ground  reaches  a 
window  12  ft.  from  the  ground.  How  far  from  the  wall  is  the  foot 
of  the  ladder  ? 

\/(13  +  12)(13  -  12)  =  V25  =  5  ft.  Ans. 

25.  The  two  legs  of  a  right  triangle  are  35  in.  and  84  in.  respec- 
tively.    Find  the  hypotenuse. 


V352  +  842  _  V8281  =  91  m.  Ans. 

26.  The  hypotenuse  of  a  right  triangle  is  61  in.,  and  one  leg  11  in. 

Find  the  other  leg. 


\/(61  +  11)(61  -  11)  =  V3600  =  60  in.  Ans. 

27.    Find  the  longest  straight  line  that  can  be  drawn  on  the  floor 
of  a  room  20  ft.  by  15  ft. 

V202  +  15»  -  V625  -  25  ft.  Ans. 


TEACHERS     EDITION. 


379 


28.   Find  the  longest  line  in  a  box  that  is  8  ft.  long,  4  ft.  wide,  1  ft. 
deep. 


V82  +  42  =  VSO.  V  VSO^  +  12  =  V81  =  9  ft.  Ans. 


Exercise  LXXVIII. 


1.   Find  the  cube  root  of  1331, 


F331(ll 
1 


3  X  102  =  300 

3(10x1)-    30 

12=      1 

331 


331 


331 


2.   Find  the  cube  root  of  1728. 


F728(12 
1 


3  X  102  =  300 

3(10x2)=    60 

22=     4 

364 


728 


728 


3.   Find  the  cube  root  of  12.167. 


12.^67(2.3 

8 


3  X  202  =  1200 

3(20x3)=    180 

32=       9 

1389 


4167 


4167 


4.   Find  the  cube  root  of  300.763. 


300.^763(6.7 
216 


3  X  602  =  10800 

3(60x7)=    1260 

72=       49 


12109 


84763 


84763 


380 


ARITHMETIC. 


5.   Find  the  cube  root  of  148,877. 


148^877(53 
125 


3  X  50»  =  7500 

3(50x3)=   450 

32=       9 

7959 


23877 


23877 


6.  Find  the  cube  root  of  2,048,383. 


2'048^3? 

1 

3x102  = 

=  300 

1048 

3(10x2)  = 

=  60 

2»  = 

-     4 

364 

728 

320383 

3  X  120*  = 

=43200 

3(120x7)  = 

=  2520 

7*  = 

=   49 

4576 

9 

320383 

7.   Find  the  cube  root  of  59.776471. 


59.^776^71  (3.91 

27 


3x30»  =  2700 

3(30x9)=   810 

9»=     81 


32776 


3591    32319 


3  X  390»  =.  456300 

3(390x1)=     1170 

1»=  1 


457471 


457471 


457471 


TEACHERS     EDITION. 


381 


8.  Find  the  cube  root  of  304957.115891. 


304^957.115^891(67.31 
216 


3  X  602  =  10800 

3(60x7)=  1260 

72=   49 


12109 


88957 


84763 


3  X  670^  =  1346700 

3(670x3)=   6030 

32=     9 


1352739 


4194115 


4058217 


3  X  67302  =  135878700 

3(6730x1)=    20190 

P=       1 


135898891 


135898891 


135898891 


9.  Find  the  cube  root  of  0.007821346625. 


0.007^821^346^625  (0.1985 
1 


3  X  102  =  300 

3(10x9)  =  270 

92=  81 

651 


6821 


5859 


3  X  1902  _  108300 

3(190x8)=  4560 

82=    64 


112924 


962346 


903392 


3  X  19802  _ 11761200 

3(1980x5)=   29700 

52=     25 


11790925 


58954625 


58954625 


382 


ARITHMETIC. 


10.  Find  the  cube  root  of  104.600290750613. 

104/600^290^750^613  (4.7117 


64 


3  X  402  =  4800 

3(40x7)=  840 

1^=     49 

5689 


40600 


39823 


3  X  470'-'  =  662700 
3  (470  X  1)  =  1410 

P  = 1 

664111 


77290 


664111 


3  X  47102  =  66552300 
3(4710x1)=   14130 

P= 1 

66566431 


113179750 


66566431 


3  X  471102  =  6658056300 
3(47110x7)=    989310 

7^  = 49 

6659045659 


46613319613 


46613319613 


11.  Find  the  cube  root  of  17,183,498,535,125. 

17^83^498^535^25  (25805 


8 


3  X  20'  =  1200 

3(20x5)=  300 

5'=  25 

1525 


9183 


7625 


3  X  250' =  187500 
3(250x8)=  6000 

8«= 64 

193564 


1558498 


1548512 


3  X  25800«  -  1996920000 
3(25800x5)-    387000 

5»« 25 

1997307025 


9986535125 


90S(ir)a5125 


TEAC^ERS     EDITION. 


383 


12.   Find  the  cube  root  of  122615.327232. 


122^615.327^232(49.68 
64 


3  X  402  =  4800 

3(40x9)  =  1080 

92=_8J_ 

5961 


58615 


53649 


3  X  4902  =  720300 
3(490x6)=     8820 

62  = 36 

729156 


4966327 


4374936 


3  X  49602  =  73804800 
3(4960x8)=      119040 

82  = 64 

73923904 


591391232 


591391232 


13.   Find  the  cube  root  of  10 ;  3| ;  8^  to  four  places  of  decimals. 

(1) 

10.000(2.1544 


3x202  = 

1200 

2000 

3(20x1)  = 

12  = 

60 
1^ 
1261  [ 

1261 

61  J 

739000 

3  X  2102  ^ 

132300 

3(210x5)  = 
52  = 

3150 
25  ^ 

135475  \ 

677375 

3175  J 

61625000 

3  X  21502  ^ 

= 13867500 

3(2150x4)  = 
42  = 

=   25800 
16  ^ 

13893316  !- 

55573264 

25816 

J 

60517360 

3  X  21542  = 

=  139191^ 

18 

53676592 

384 


ARITHMETIC. 


^Sf 


(2) 
2     2 


3  X  3002  _  270000 
3(300x7)=  6300 

72  = 49 

276349 
6349 


3  X  30702  =  28274700 
3(3070x2)=   18420 

22  = 4 

28293124 

18424 . 

3  X  30722  =  28311552 


29.000(3.0722 

27 


2000000 


1934443 


65557000 


56586248 


89707520 
56623104 


(3) 


8.333^333(2.0274 
8 


3  X  2002  =  120000 
3(200x2)=  1200 

22  = 4-) 

121204  [ 
1204  J 
3  X  20202  =  12241200 
3(2020x7)=   42420 

7«  = 49 

122836(: 
42469 


333333 


242408 


569  [ 
169  J 


3  x  20272  =  12326187 


90925333 


85985669 


49396643 
44304748 


TEACHERS     EDITION. 


385 


14.  Find  the  cube  root  of  5  ;  f  ;  7|- ;  f  to  four  places  of  decimals. 

(1) 

5.000(1.7099 
1 

3  X  10'^  =  300 

4000 

3(10x7)^-^210 

72=  49  ^ 
559  [ 

3913 

259  J 

87000000 

3  X  1700*-*  =  8670000 

3(1700x9)=  45900 

92  =     81  ^ 

8715981  [ 

78443829 

45981  J 

85561710 

3  X  17092  =  8762043 

78858387 

(2) 

0.555^555(0.8221 

512 

3  X  802  _  19200 

43555 

3(80x2)=  480 

22  =    4  ^ 

19684  [ 

39368 

484  i 

41875 

3  X  822  =  20172 

40344 

15315 

(3) 

7.600(1.966 
1 

3x102=  300 

6600 

3(10x9)=  270 

92  =  81  -J 

_5859  _ 

651  ■ 

741000 

351  J 

3  X  1902  =  108300 

3(190x6)=  3420 

62  =    36  -| 

111756  ■ 

670536 

3456  i 

704640 

3  X  1962  =  115248 

691488 

386 


ARITHMETIC. 


(4) 


0.750^000^000(0.9085 
729 


S  X  9002  =  2430000 
3(900x8)=      21600 

82  = 64  ^ 

2451664  [ 
21664  J 
2473392 


21000000 


19613312 


13866880 
12366960 


15.   Find  the  entire  surface   of  a  cube   the  volume  of  which  is 
14  cu.  ft.  705.088  cu.  in. 

14  cu.  ft.  705.088  cu.  in. 
X1728 


24192  cu.  in. 
705.088 

24897.088  cu.  in 


24^897.088  (29.2  in. 


3  X  202  =  1200 

3(20x9)=   540 

92  =  _81  -| 

1821  [ 

621  J 

3  X  2902  _  252300 

3(290x2)=      1740 

2»- 4 

254044 


16897 


16389 


508088 


508088 


29.2 
X29.2 

852.64  sq.  in.  in  each  face. 

6 

144)5115.84  sq.  in. 

35  sq.  ft  75.84  sq.  in. 


teachers'  edition.  387 


Exercise   LXXIX. 

1.  If  the  diameter  of  the  moon  be  reckoned  at  2000  mi.,  and  that 
of  the  earth  at  8000  mi.,  find  the  ratio  of  the  surface  of  the  moon  to 
that  of  the  earth.  Also,  find  the  ratio  of  the  volume  of  the  moon  to 
that  of  the  earth. 

20002 .  80002  =  12 .  42  =  1 .  16.  (1)  Ans. 
20003  .  80003  =  13  :  43  =  1 :  64.  (2)  Ans. 

2.  If  the  diameter  of  the  earth  be  reckoned  at  8000  mi.,  that  of 
Jupiter  at  84,000  mi.,  and  that  of  the  sun  at  880,000  mi.,  find  the 
ratios  of  their  volumes. 

80003 .  840003 .  8800003  _  23 ;  213 :  2203  =  8  :  9261  :  10648000.  Ans. 

3.  If  the  diameters  of  two  circles  be  20  in.  and  40  in.  respectively, 
find  the  ratio  of  their  circumferences  and  the  ratio  of  their  surfaces. 

20  :  40  =  1  :  2. 
202 :  402  =  12 :  22  =  1 :  4.  Ans. 

4.  If  the  areas  of  two  circles  be  8000  sq.  in.  and  36,000  sq.  in. 
respectively,  find  the  ratio  of  their  diameters  to  the  nearest  thou- 
sandth of  an  inch. 

a/SOOO  :  V36000  -=  Vi :  \/l8  =  2  :  4.242  =  1 :  2.121.  Ans. 


5.  If  the  volumes  of  two  spheres  be  100  cu.  in.  and  1000  cu.  in. 
respectively,  find  the  ratio  of  their  diameters  to  the  nearest  thou- 
sandth of  an  inch. 

\/lOO:  v^IOOO  =  v'l :  v/IO  =  1  :  2.154.  .4ns. 


6.  If  two  stacks  of  hay  of  the  same  shape  contain  4  t.  6  cwt.  and 
1  t.  8  cwt.  respectively,  find  the  ratio  of  their  heights. 

v^86  :  \/28  =  4.414  :  3.037  =  1 :  0.688.  Ans. 


388  ARITHMETIC. 


7.   If  an  ox  7  ft.  in  girth  weigh  1500  lbs.,  what  will  be  the  girth 
of  a  similar  ox  weighing  2500  lbs.? 


v^l500  :  V^2500  :  :  7  ft. :  what? 
^i:  ^^/fUJ:  :7  ft.:  what? 
1:1.186:  :7  ft.:  what? 
1  :  1.186  :  :  7  ft.  :  8.3  ft.  Ans. 


8.  TLe  surface  of  a  pyramid  is  560  sq.  in.  What  is  the  surface  of 
a  similar  pyramid  whose  volume  is  27  times  as  great? 

^:^^  =  1:3. 

P:32:  :560sq.  in.:  what? 

1:9::  560  sq.  in. :  5040  sq.  in.  Am. 

9.  The  volume  of  a  pyramid  is  1331  cu.  in.  What  is  the  volume 
of  a  similar  pyramid  whose  surface  is  4  times  as  great? 

Vl :  \/4  =  1 :  2. 

P:  23::  1331  CU.  in.  :  what? 

1:8::  1331  cu.  in. :  10648  cu.  in.  Arus. 

10.  If  a  well-proportioned  man  5  ft.  10  in.  high  weigh  160  lbs., 
what  should  a  man  6  ft.  high  weigh,  to  the  nearest  tenth  of  a  pound  ? 
What  should  be  the  height,  to  the  nearest  tenth  of  an  inch,  of  a  man 
weighing  210  lbs.  ? 

5  ft.  10  in.  =  70  in.        6  ft.  =  72  in. 

703 :  723.  .  leoibs. :  what? 

343000 :  373248  : :  160  lbs. :  174.11bs.  (1)  Ans. 

Vm-.  v/210::70in.  :what? 

5.43  :  5.95  :  :  70  in.  :  76.6  m. 

76.6  in.  =6  ft.  4.6  in.  (2)  Ans. 

11.  A  three-gallon  jug  and  a  one-gallon  jug  are  of  the  same  shape 
What,  to  the  nearest  thousandth,  is  the  ratio  of  their  diameters  ? 

v^  :  v^  =  1.443  :  1  =  1  :  0.693.  Ana. 


teachers'  edition.  389 

12.  Two  hills  have  exactly  the  same  shape;  one  is  900  ft.  high, 
the  other  1200  ft.  Find  the  ratio  of  their  surfaces,  and  also  the  ratio 
of  their  volumea. 

9002 .  12002  _  32 .  42  _  9  .  ig    (1)  j^^g 
9003  .  12003  =  33 :  43  =  27  :  64.  (2)  Ans. 

13.  A  ball  3  in.  in  diameter  weighs  4  lbs. ;  another  ball  of  the 
same  metal  weighs  9  lbs.  Find  the  diameter  of  the  second  ball  to 
the  nearest  thousandth  of  an  inch. 

v/4:  v^9::3in.:what? 

1.587  :  2.080  :  :  3  in. :  3.931  in.  Ans. 

14.  If  Apollo's  altar  were  a  perfect  cube  10  ft.  on  a  side,  what,  to 
the  nearest  hundredth  of  an  inch,  would  be  the  dimensions  of  a  new 
cubical  altar  containing  twice  as  much  stone? 

10  X  10  X  10=  1000  cu.  ft. 
2  X  1000  =  2000  cu.  ft. 
\/l000  :  v^2000  :  :  10  :  what? 
10  :  12.599  :  :  10  :  12.599  ft. 
12.599  ft.  =  12  ft.  7.19  in.  Ans. 

15.  A  man  standing  40  ft.  from  a  building  24  ft.  wide  observed 
that,  when  he  closed  one  eye,  the  width  of  the  building  hid  from 
view  90  rods  of  fence  which  was  parallel  to  the  width  of  the  build- 
ing.    Find  the  distance  from  the  eye  of  the  observer  to  the  fence. 

24:40::90rds.  :what? 

24  :  40  :  :  90  rds.  :  150  rds.  Ans. 

16.  A  bushel  measure  and  a  peck  measure  are  of  the  same  shape. 
Find  the  ratio  of  their  heights. 

1  bushel  =  4  pecks. 

\/4  :  v^I  =  1.587  :  1  =  1 :  0.63.  Ans. 


390 


ARITHMETIC. 


EXEKGISE     LXXX. 


Given  :  log  2  =  0.3010  ;  log  3  =  0.4771 ;  Ibg  5  =  0.6990 ;  log 
7  =  0.8451. 

Find  the  logarithms  of  the  following  numbers  by  resolving  the 
numbers  into  factors,  and  taking  the  sum  of  the  logarithms  of  the 
factors. 


log   6  =  log  (2x3) 

=  log2  +  log  3. 
log   2  =  0.3010 
log   3  =  0.4771 

0.7781.  Am. 


2. 

log  15  =  log  (3  X  5) 

=  log  3+ log  5. 
log   3  =  0.4771 
log  5  =  0.6990 

1.1761.  Am. 


log  21  =  log  (3x7) 

=  log  3  +  log  7. 
log   3  =  0.4771 
log  7  =  0.8451 

1.3222.  Am. 


log  14  =  log  (2  X  7) 

=  log  2  +  log  7. 
log   2  =  0.3010 
log   7  =  0.8451 

1.1461.  Am. 


5. 

log  35  =  log  (5x7) 

=  log  5  +  log  7. 
log   5  =  0.6990 
log   7  =  0.8451 


1.5441. 

Am. 

6. 

log 

9  = 

=  log(3x 

3) 

= 

=  log3  + 

logs. 

log 

3  = 

=  0.4771 

log 

3  = 

=  0.4771 

0.9542. 

Am. 

7. 

log 

8  = 

=  log(2x 

2x2) 

= 

=  log2  + 

log  2  + log  2. 

log 

2  = 

=  0.3010 

log 

2 

=  0.3010 

log 

2  = 

=  0.3010 

0.9030. 

Am. 

8. 

log 

49  = 

=  log  (7x7) 

=  log7  + 

log  7. 

log 

7 

=  0.8451 

log 

7 

=  0.8451 

1.6902.  Am. 


TEACHERS     EDITION. 


391 


9. 

log   25  =  log  (5x5) 

=  log  5  +  log  5, 
log     5  =  0.6990 
log     5  =  0.6990 

1.3980.  Ans. 


10. 

log   30  =  log (2x3x5) 

=  log  2  +  log  3  +  log  5. 
log     2  =  0.3010 
log     3  =  0.4771 
log     5  =  0.6990 

1.4771.  Ans. 


11. 

log   42  =  log(2x3x7) 

=  log  2  +  log  3  +  log  7. 
log     2  =  0.3010 
log     3  =  0.4771 
log     7  =  0.8451 

1.6232.  Ans. 


12. 

log  420  =  log  (2x2x3x5x7) 
=  log  2  +  log  2  +  log  3 
+  log  5  +  log  7. 
log      2  =  0.3010 
log     2  =  0.3010 
log     3  =  0.4771 
log     5  =  0.6990 
log      7  =  0.8451 

2.6232.  Ans. 


I  13. 

log  12  =  log  (2  X  2  X  3) 

=  log2  +  log2  +  log3. 
log    2=0.3010 
log    2  =  0.3010 
loR    3  =  0.4771 


1.0791.  Ans. 


14. 


log  60  =  log(2x  2x3x5) 
=  log  2  +  log  2  +  k^g  3 


log  2  =  0.3010 

log  2  =  0.3010 

log  3  =  0.4771 

log  5  =  0.6990 


+  iogo. 


1.7781.  Ans. 

15. 

log  75  =  log(3x5x5) 

=  log  3  +  log  5  +  log  5. 
log    3  =  0.4771 
log    5  =  0.6990 
log    5  =  0.6990 

1.8751.  Ans. 

16. 

log7.5  =  log(5x  5x3x0.1) 
=  log  5  +  log  5  +  log  3 

+  log  0.1. 
log    5  =  0.6990 
log    5  =  0.6990 
log    3  =  0.4771 
log0.1  =  9.0000 -10 

0.8751.  Ans. 


392 


ARITHMETIC. 


17. 

log  0.021  =  log(3x  7x0.001) 
=  log  3  +  log  7 

+  log  0.001. 
log         3  =  0.4771 
log  7  =  0.8451 

log  0.001  =  7.0000-10 

8.3222  -  10.  Ans. 


18. 

log     0.35  =  log  (5  X  7  X  0.01) 
=  log  5  +  log  7 

+  log  0.01. 
log  5  =  0.6990 

log  7  =  0.8451 

log     0.01  =  8.0000  -  10 

9.5441-10.  Ans. 


19. 

log  0.0035  =  log  (5  X  7  X  0.0001) 
=  log5  +  log7 

+  log  0.0001. 
log         5  =  0.6990 
log  7  =  0.8451 

log  0.0001  =  6.0000 -10 

7.5441  -  10.  Am. 

20 

log   0.004  =  log  (2  X  2  X  0.001) 
=  log2  +  log2 

+  log  0.001. 
log  2  =  0.3010 

log  2  =  0.3010 

log  0.001  =  7.0000-10 

7.6020-10.  A71S. 


21. 

log  0.05  =  log  (5  X  0.01) 

=  log  5  +  log  0.01. 
log      5  =  0.6990 
log  0.01  =  8.0000  -  10 

8.6990  -  10.  Ans. 


22. 

log  12.5  =  log  (5  X  5  X  5  X  0.1) 
=  log  5  +  log  5  +  log  5 
+  log  0.1. 
log      5  =  0.6990 
log      5  =  0.6990 
log      5  =  0.6990 
log  0.1  =  9.0000-10 

1.0970.  Ans. 

23. 

logl.25  =  log(5x5x  5x0.01) 
=  log  5  +  log  5  +  log  5 
+  log  0.01. 
log      5  =  0.6990 
log      5  =  0.6990 
log      5  =  0.6990 
log0.01=  8.0000- 10 

0.0970.  Ans. 

24. 

log  37.5  =  log  (3  X  5  X  5  X  5  X  0.1) 
=  log  3  +  log  5  +  log  5 
+  log  5  +  log  0.1. 
log      3  =  0.4771 
log      5  =  0.6990 
log      5  =  0.6990 
log      5  =  0.6990 
log   0.1  =  9.0000  -  10 

1.5741.  Ans. 


TEACHERS     EDITION. 


393 


25. 

log      2.1  =  log(3x  7x0.1) 

=  log3s  Iog7  +  log0.1. 
log        3  =  0.4771 
log        7  =  0.8451 
log     0.1  =  9.0000  -  10 


0.3222.  Ans. 


26. 


log 


16  =  log  (2  X  2  X  2  X  2) 
-  log  2  +  log  2  +  log  2 
+  log2 
log        2  =  0.3010 
log         2  =  0.3010 
log         2  =  0.3010 
log        2  =  0.3010 

1.2040.  Ans. 

27. 

log  0.056  =  log  (2x2x2x7x0.001) 
=  log2  +  log2  +  log2 
+  log  7+ log  0.001. 
log        2  -  0.3010 
log        2  =  0.3010 
log        2  =  0.3010 
log        7  =  0.8451 
log  0.001  =  7.0000 -10 

8.7481  -  10.  Ans. 

28. 

log    0.63  =  log  (3  X  3  X  7  X  0.01) 
=  log3  +  log  3+ log  7 
+  log  0.01. 
log        3  =  0.4771 
log        3  =  0.4771 
log        7  =  0.8451 
log  0.01  =  8.0000  -  10 

9.7993-10.  .4ns. 


29. 


log 


1.75  =  log(5x  5x7x0.01) 
=  log  5  +  log  5  +  log  7 
+  log  0.01. 
log  5  =  0.6990 

log  5  =  0.6990 

log  7  =  0.8451 

log     0.01  =  8.0000  -  10 

0.2431.  Ans. 


30. 

105  =  log  (3  X  5  X  7) 


log 

=  log3  +  log  5  +log7. 
log  3  =  0.4771 

log  5  =  0.6990 

log  7  =  0.8451 

2.0212.  Ans. 

31. 

log  0.0105  =  log  (3x5x7x0.0001) 
=  log  3  +  log  5  +  log  7 
+  log  0.0001. 
log  3  =  0.4771 

log  5  =  0.6990 

log  7  =  0.8451 

log  0.0001  =  6.0000 -10 

8.0212-10.  Ans. 

32. 

log      1.05  =  log(3x  5x7x0.01) 
=  log3  +  log5  +  log7 
+  log  0.01. 
log  3  =  0.4771 

log  5  =  0.6990 

log  7  =  0.8451 

log     0.01  =  8.0000  -  10 

0.0212.  Ans. 


394  ARITHMETIC. 


Exercise  LXXXI. 

Given  :    log  2  =  0.3010  ;    log  3  =  0.4771 ;    log  5  =  0.6990 ;    log  7 
=  0.8451. 

Find  logarithms  of  the  following  : 

1.  log  2' ==  3  X  log  2  =  3  X  0.3010  =  0.9030.  Am. 

2.  log  52  =  2  X  log  5  =  2  X  0.6990  =  1.3980.  Am. 

3.  log  7*  =  4  X  log  7  =  4  X  0.8451  =  3.3804.  Am. 

4.  log  38  =  8  X  log  3  =  8  X  0.4771  =  3.8168.  Am. 

5.  log  73  =  3  X  log  7  =  3  X  0.8451  =  2.5353.  Am. 

6.  log5*  =  5xlog5  =  5x  0.6990  =  3.4950.  Ans. 

7.  log  2s    =    I  of  log  2  =    A  of  0.3010  =  0.1003.  Am. 

8.  log  5^-    =    ^  of  log  5=    ^  of  0.6990  =  0.3495.  ^7w. 

9.  log3s    =    ^  of  log  3=    i  of  0.4771  =  0.0596.  ^?w. 

10.  log  7^    =    ^  of  log  7  =    i  of  0.8451  =  0.1690.  Am. 

11.  log5i    =    ^  of  log  5=    ^  of  0.6990  =  0.1398.  ^rw. 

12.  log  71^  =  T^r  of  log  7  =  3»y  of  0.8451  =  0.0768.  Am. 

13.  log  2I    =    f  of  log  2  =    I  of  0.3010  =  0.2258.  Am. 

14.  log5t    =    I  of  log  5=    f  of  0.6990  =  0.4660.  ^TM. 

15.  log  3^    =    f  of  log  3=    f  of  0.4771  =  0.2045.  Am. 

16.  log  7^    -    ^  of  log  7=    ^  of  0.8451=  0.2415.  ^rw. 

17.  log  5*    =    f  of  log  5=    I  of  0.6990  =  1.1650.  Am. 

18.  log  :>  u  =  i^r  of  log  3  =  x'r  of  -.4771  =  0.390-1.  Am. 


TEACHERS     EDITION. 


395 


19.  log  7^    =    I  of  log  7=    I  of  0.8451  =  2.9579.  ^w«. 

20.  logSt    =    I  of  log  3-    -f  of  0.4771  -  0.6361.  Ans. 

21.  logSt    =    I  of  log  5=    I  of  0.6990  =  2.4465.  Ans, 

22.  log  2'T  =  -If  of  log  2  =  -If  of  0.3010  -  0.4730.  Ans. 

23.  log  51    =    f  of  log  5=    f  of  0.6990  =  0.5243.  ^ns. 

24.  log  7-V-  =  If  of  log  7  =  -V-  of  0.8451  =  1.3280.  yins. 

26.   log  2li  =  I  of  log  21  =  I  of  log  (3  x  7) 

=  I  of  (0.4771  +  0.8451) 

=  I  of  1.3222  =  1.1569.  Ans. 


Exercise  LXXXII. 

^3  =  0.4771;    log  5  =  0.6990;    log  7 


Given  :    log  2  =  0.3010 ; 
=  0.8451. 

Find  logarithms  for  the  following  quotients : 


^og  f  ^  ^°g  2-|-colog5— 10. 

log  2  =  0.3010 

colog5-10  =  9.3010-10 

9.6020-10.  Ans. 

2. 

log  f  =  log  2+colog  7-10. 

log  2  =  0.3010 

colog7-10  =  9.1549-10 

9.4559-10.  Ans. 


log  I  =  log  3  4-colog  5  - 10. 

log  3  =  0.4771 

colog5-10  =  9.3010-10 

9.7781  -  10.  Am. 


log  f  =  log  3  -Fcolog  7-10. 

log  3  =  0.4771 

colog  7  -  10  =  9.1549  -  10 

9.6320-10.  Ans. 


5. 

log  f  =  log5-i-colog7— 10. 

log  5  =  0.6990 

colog  7 -10  =  9.1549 -10 

9.8539  -  10.  Ans. 


log  ^  =  log  7 +colog  5  - 10. 

log  7  =  0.8451 

cologS- 10  =  9.3010 -10 

0.1461.  Ans. 


396 


ARITHMETIC. 


log  f  ==  log  5  +  colog  3—10. 

log  5-0.6990 

colog  3  -  10  =  9.5229  -  10 

0.2219.  Am. 

8. 

log  I  =  log  5 + colog  2—10. 

log  5  =  0.6990 

colog  2  -  10  =  9.6990  -  10 

0.3980.  Ans. 

9. 

log  I  =  log  7 + colog  3-10. 

log  7  =  0.8451 

colog  3  -  10  =  9.5229  -  10 

0.3680.  Am. 


10. 

log  1  =  log  7  +  colog  2-10. 

log  7  =  0.8451 

colog  2 -10  =  9.8990 -10 

0.5441.  Am. 

11. 

log  f  =  log  3  +  colog  2-10. 

log  3  =  0.4771 

colog  2  -  10  =  9.6990  -  10 

0.1761.  Am. 
12. 
log  -^  =  Iog7+colog0.5-10. 

log  7=   0.8451 

colog0.5-10=  10.3010 -10 

1.1461.  Am. 


log 


0.05 
3 
log         0.05  = 
colog  3  -  10  = 


13. 

=  log  0.05  + colog  3 -10. 

8.6990  -  10 
9.5229  -  10 


log 


0.005 


8.2219  -  10.  Am. 


14. 

log  0.005  + colog  2 -10. 


log      0.005  =  7.6990  -  10 
colog  2  -  10  =  9.6990  -  10 

7.3980  - 10.  Am. 


log 


0.07 
5 


16 

log  0.07  + colog  5 -10. 


log         0.07  =  8.8451  -  10 
colog  5  -  10  =  9.3010  -  10 


8.1461  -  10.  Am. 


teachers'  edition.  397 


16. 

log  ^  =  log  5  +  colog  0.07  -  10. 

log  5=    0.6990 

colog   0.07-10  =  11.1549-10 

1.8539.  Ans. 

17. 

log  — ^  =  log  3  +  colog  0  007  -  10. 

°  0.007 

log  3=   0.4771 

colog  0.007  -  10  =  12.1549  -  10 

2.6320.  Ans. 

18. 

log  ^^  =  log  0.003  +  colog  7  -  10. 

log  0.003  =  7.4771  -  10 

colog        7  -  10  =  9.1549  -  10 

6.6320  -  10.  Ans. 

19. 

log  -^:^  =  log  0.05  +  colog  0.003  -  10. 

^  0.003        6^6 

log  0.05=    8.6990-10 

colog  0.003  -  10  =  12.5229  -  10 

1.2219.  Ans. 

20. 

log  M^  =  log  0.007  +  colog  0.02  -  10. 

log  0.007=    7.8451-10 

colog   0.02  -  10  =  11.6990  -  10 

9.5441  - 10.  Ans. 


398  ARITHMETIC. 


21. 

log  -M2  =  log  0.02  +  colog  0.007  -  10. 

^  0.007        ^  ^ 

log  0.02=    8.3010-10 

colog  0.007  -  10  =  12.1549-10 

0.4559.  Am. 


22. 

log  2i^  =  log  0.005  +  colog  0.07  -  10. 

log  0.005=    7.6990-10 

colog  0.07-10=11.1549-10 

8.8539-10.  Am. 

23. 

log  2:^  =  log  0.03  +  colog  7  -  10, 

log  0.03  =  8.4771  -  10 

colog        7  -  10  =  9.1549  -  10 

7.6320-10.  Ans. 

24. 

log  ^^991  =  log  0.0007  +  colog  0.2  -  10. 

log         0.0007    =   6.8451-10 
colog     0.2  -  10  =  10.6990  -  10 

7.5441  - 10.  Am. 

25. 

log  2:^  -  log  0.02«  +  colog  3»  - 10. 

o 

log  0.02'^  =  6.6020  -  10 

colog      3»  -  10  =  8.5687  -  10 

5.1707  - 10.  Am. 


teachers'  edition.  399 

26. 

log  ^  =  log33  +  colog0.02^-10. 

log  33=    1.4313 

colog   0.022  -  10  =  13.3980  -  10 

4.8293.  Ans. 

27. 

log  ^  =  log  73  +  colog  0.022  _  10. 

log  73=    2.5353 

colog  0.022-10=13.3980-10 

5.9333.  Ans. 

28. 

log  -Mli  _  log  0.073  +  colog  0.0033  _  10. 

^  0.0033  &  -r  & 

log  0.073=    6.5353-10 

colog  0.0033  -  10  =  17.5687  -  10 

4.1040.  Ans. 

29. 

log  2:^  =  log  0.0052  ^  colog  73  -  10. 

log  0.0052  _  5  3980  _  10 

colog        73  -  10  =  7.4647  -  10 

2.8627  - 10.  Ans. 

30. 

log  -^  =  log  73  +  colog  0.0052  _  10. 

^  0.0052        &  & 

log  73=    2.5353 

colog  0.0052  -  10  =  14.6020  -  10 

7.1373.  Ans. 


400  ARITHMETIC. 


Exercise  LXXXIII. 
Find  logarithms  of  the  following  numbers. 

1. 

log  70  =  1.8451.  Am. 

2. 

log  101  =  2.0043.  Ans. 

3. 

log  333  =  2.5224.  Am. 

4. 

log    3491  =  3.5428  +  (yV  of  13)    =  3.5429.  Am. 

5. 

log    1866  =  3.2695  +  {^  of  23)   =  3.2709.  Am. 

6. 

log    6897  =  3.8382  +  (tV  of  6)     =3.8386.  Am. 

7. 

log    9901  =  3.9956  +  (yV  of  5)     =  3.9957.  Am. 

8. 

log    4389  =  3.6415  +  {j%  of  10)    =  3.6424.  Am. 

9. 

log    nil  =  3.0453  +  (tV  of  39)   =  3.0457.  Avs. 

10. 

log  58343  =  4.7657  +  {j%%  of  7)   =  4.7660.  Am. 

11. 

log  77860  =  4,8910  +  {^%  of  5)   =  4.8913.  Am. 

12. 

log  30127  =  4.4786  +  {^%\  of  14)  =  4.4790.  Am. 

13. 

log  730.84  =  2.8633  +  {^Ss  of  6)   -  2.8638.  Am, 


teachers'  edition.  401 

14. 

log  0.008765  =  7.9425  +  {j%  of  5)  -  10  =  7.9428  -  10.  Am. 

15. 

log     8.0808  =  0.9074  +  (yf  „  of  5)  =  0.9074.  Am. 

16. 

log     5.0009  =  0.6990  +  {j^  of  8)  =  0.6991.  Am. 

17. 

log     0.3769  =  9.5752  +  {j\  of  11)  -  10  =  9.5762  -  10.  Am. 

18. 

log  0.070707  =  8.8494  +  (j^^  of  6)  -  10  =  8.8494  -  10.  Am. 

19. 

log  0.03723  =  8.5705  +  {j%  of  12)  -  10  =  8.5709  -  10.  Am. 

20. 

log     98.871  =  1.9948  +  (^5^^  of  4)  =  1.9951.  Am. 

21.  Find  antilog  3.9017. 

The  number  corresponding  to  the  mantissa  9015  is  7970. 
The  number  corresponding  to  the  mantissa  9020  is  7980. 
The  difference  between  these  numbers  is  10, 
and  7970  +  f  of  10  =  7974.  Am. 

22.  Find  antilog  1.2076. 

The  number  corresponding  to  the  mantissa  2068  is  1610. 
The  number  corresponding  to  the  mantissa  2095  is  1620. 
The  difference  between  these  numbers  is  10, 

and  1610  +  ^\  of  10  =  1613. 

Therefore,  the  number  required  is  16.13.  Ans. 

23.  Find  antilog  0.4442. 

The  number  corresponding  to  the  mantissa  4440  is  2780. 
The  number  corresponding  to  the  mantissa  4456  is  2790. 
The  difference  between  these  numbers  is  10, 

and  2780  +  y2^  of  10  =  2781. 

Therefore,  the  number  required  is  2.781.  Am. 


402  ARITHMETIC. 


24.  Find  antilog  1.0090. 

The  number  corre8{)onding  to  the  mantifsa  0086  is  1020. 
The  number  corresponding  to  the  mantissa  0128  i»  lOlJO. 
The  difference  between  these  numbers  is  10, 

and  1020 +  5\  of  10  =  1021. 

Thwefore,  the  number  required  is  10.21.  Ans. 

25.  Find  antilog  2.9850. 

The  number  corresponding  to  the  mantissa  9850  is  9660. 
Therefore,  the  number  required  is  966.  Ans. 

26.  Find  antilog  4.5388. 

The  number  corresponding  to  the  mantissa  5378  is  3450. 
The  number  corresponding  to  the  mantipsa  5391  is  3460. 
The  difference  between  the  numbers  is  10, 
and  3450  +  |f  of  10  =  3458. 

Therefore,  the  number  required  is  34,580.  Ans. 

27.  Find  antilog  0.8550. 

The  number  corresponding  to  the  mantissa  8549  is  7160. 
The  number  corresponding  to  the  mantissa  8555  is  7170. 
The  difference  between  these  number  is  10, 
and  7160  +  ^  of  10  =  7162. 

Therefore,  the  number  required  is  7.162.  Aiu, 

28.  Find  antilog  9.9992  -  10. 

The  number  corresponding  to  the  mantissa  9991  is  9980. 
The  number  corresponding  to  the  mantissa  9996  is  9990. 
The  difference  between  these  numbers  is  10, 

and  9980  +  ^  of  10  =  9982. 

Therefore,  the  number  required  is  0.9982.  Ans. 

29.  Find  antilog  8.7324  -  10. 

The  number  corrosj)onding  to  the  mantissa  7324  is  5400, 
Therefore,  the  number  required  is  0.0540.  Am. 


teachers'  edition.  403 


30.  Find  antilog  9.5555  -  10. 

Tlio  number  corresponding  to  the  mantissa  5551  is  3590. 
The  number  corresponding  to  the  mantissa  5563  is  3600. 
The  difference  between  these  numbers  is  10, 
and  3590  +  ^^  of  10  =  3593. 

Therefore,  the  number  required  is  0.3593,  Ans. 

31.  Find  antilog  6.0216  -  10. 

The  number  corresponding  to  the  mantissa  0212  is  1050. 
The  number  corresponding  to  the  mantissa  0253  is  1060. 
The  difference  between  these  numbers  is  10, 
and  1050 +  5^  of  10  =  1051. 

Therefore,  the  number  required  is  0.0001051.  Ans. 

32.  Find  antilog  7.0080  -  10. 

The  number  corresponding  to  the  mantissa  0043  is  1010. 
The  number  corresponding  to  the  mantissa  0086  is  1020. 
The  difference  between  these  numbers  is  10, 
and  1010 +  11  of  10  =  1019. 

Therefore,  the  number  required  is  0.001019.  Ans. 

33.  Find  by  logarithms  the  product  948.22  x  0.4387. 

log   948.22  =  2.9769 
log  0.4387  =  9.6422  -  10 

2.6191  =  log  416.  Ans. 

34.  Find  by  logarithms  the  product  1.9704  x  0.0786. 

log    1.9704  =  0.2946 
log   0.0786  =  8.8954-10 

9. 1900 -10  =  log  0.1 549.  ^ns. 

35.  Find  by  logarithms  the  product  380.25  x  0.00673. 

log   380.25  =  2.5801 

log  0.00673  =  7.8280 -10 


0.4081  =  log  2.559.  Am. 


404  ARITHMETIC. 


36.  Find  by  logarithms  the  product  216.21  X  0.76312. 

log   216.21  =  2.3349 
log  0.76312  =  9.8826 -10 

2.2175  =  log  165.  Ans. 

37.  Find  by  logarithms  the  product  0.56127  X  1.2312. 

log  0.56127  =  9.7492 -10 
log   1.2312  =  0.0903 

9.8395  -  10  =  log  0.691 .  Ans. 

38.  Find  by  logarithms  the  product  0.86311  X  56.371. 

Iog0.86311  =  9.9361 -10 
log  56.371  =  1.7511 

1.6872  =  log  48.67.  Ans. 

39.  Find  by  logarithms  the  product  59.795  x  0.7955. 

log  59.795  =  1.7767 
log  0.7955  =  9.9007-10 

1.6774  =47.58.  Ans. 

40.  Find  by  logarithms  the  product  270.05  X  0.0087. 

log   270.05  =  2.4315 
log  0.0087  =  7.9395-10 

0.3710  =  log  2.349.  Ans. 


41.  Find  by  logarithms  the  product  11.163  x  0.3333. 

log   11.163  =  1.0478 
log  0.3333  =  9.5228-10 

0.5706  =  log  3.721.  Am. 

42.  Find  by  logarithms  the  product  777.78  x  0.0787. 

log   777.78  =  2.8909 
log   0.0787  =  8.8960  -  10 

1.7869  =  log  61.21.  Ans. 


teachers'  edition.  405 

43.  Find  by  logarithms  the  product  2.6537  X  0.2313. 

log   2.6537  =  0.4238 

log  0.2313  =  9.3642  -  10  " 

9.7880  -  10  =  log  0.6137.  Ans. 

44.  Find  by  logarithms  the  product  37.587  X  12.371. 

log  37.587  =  1.5750 
log   12.371=1.0924 

2.6674  =  log  464.9.  Ans. 

45.  Find  by  logarithms  the  product  89.313  x  2.3781. 

log   89.313  =  1.9510 
log   2.3781  =  0.3762 

2.3272  =  log  212.4.  ^ns. 

46.  Find  by  logarithms  the  product  9.1765  X  0.00089.  X 

log   9.1765  =  0.9627 
log  0.00089  =  6.9494 -10 


7.9121  - 
e  quotient  ot^fi-JOT. 

log     56.407=1.7513 
colog  13.045  =  8.8846 - 

-10  =  log  0.008168.  Ans. 
-10 

0.6359 

,.     ,    .857.06 
e  Quotient  oi 

=  log  4.324.  Ans. 

3079.8 
log     857.06  =  2.9330 
colog  3079.8  =  6.5114 -10 


49.    Find  the  quotient  of 


9.4444  -  10  =  log  0.2783.  Ans. 
0.9387 


598.6 

log     0.9387  =  9.9726  -  10 
colog    598.6  =  7.2229-10 


7.1955  -  10  =  log  0.001569.  Ans. 


406  ARITHMETIC. 


50.    Find  the  quotient  of  -^2^. 
^  0.7891 

log       3069=    3.4870 
colog0.7891  =  10.1028 -10 


3.5898         =  log  3889.  Am. 


51.   Find  the  quotient  of  ^^-^^XQQ^^^. 
^  93.08  X  98.071 

log       75.46  =  1.8777 
log     0.0765  =  8.8837  -  10 
colog   93.08  =  8.0312-10 
colog  98.071  =  8.0084  -  10 


6.8010-10    =  log  0.0006324.  ^rw. 


52.    Fmd  the  quotient  of  98x537x0.0079, 
^  67309  X  0.0947 

log  98=    1.9912 

log  537=    2.7300 

log  0.0079=  7.8976-10 
colog  67309=  5.1719-10 
colog  0.0947  =  11.0237  -  10 


8.8144  -  10  =  log  0.06523.  Am. 


53.    Fmd  the  quotient  of  314X7-18X8132 


^  H""^' 

'^ 519  X  827  X  3.215 

log 

314  =  2.4969 

log 

7.18  =  0.8561 

log 

8132  =  3.9102 

colog 

519  =  7.2848  - 10 

colog 

827  =  7.0825  -  10 

colog 

3.215  =  9.4928  -  10 

1.1233  =  log  13.28.  Am. 


teachers'  edition.  407 

KA     T?-   A  ,\.  ^-     ,    r  212x2.16x8002 

54.  Find  the  quotient  of ^ 

^  536  X  351  X  7.256 

log  212  =  2.3263 

log  2.16  =  0.3345 

log  8002  =  3.9032 

colog  536  -  7.2708  - 10 

colog  351  =  7.4547  -  10 

dolog  7.256  =  9.1393-10 

0.4288        =  log  2.684.  Ans. 

55.  Find  by  logarithms  5.06'. 

3  log  5.06  =  3  X  0.7042  =  2.1126 
2.1126  =  log  129.6.  Ans. 

56.  Find  by  logarithms  2.50P. 

51og2.501  =5x0.3981=  1.9905 
1.9905  =  log  97.84.  Ans. 

57.  Find  by  logarithms  1.7161 

7  log  1.716  =  7  X  0.2345  =  1.6415 
1.6415  =  log  43.8.  Ans. 

58.  Find  by  logarithms  1.178i». 

10  log  1.178  =  10  X  0.0712  =  0.7120 
0.7120  =  log  5.153.  Ans. 

59.  Find  by  logarithms  0.7685«. 

6  log  0.7685  =  6  X  (9.8857  -  10)  =  9.3142  -  10 
9.3142  -  10  =  log  0.2061.  Ans. 

60.  Find  by  logarithms  0.96118. 

8  log  0.9611  =  8  X  (9.9828  -  10)  =  9.8624  -  10 
9.8624  -  10  =  log  0.7285.  Ans. 


408  ARITHMETIC. 


61.  Find  by  logarithms  0.02312. 

2  log  0.0231  =  2  X  (8.3636  -  10)  =  6.7272  -  10 
6.7272  -  10  =  log  0.0005336.  Ans. 

62.  Find  by  logarithms  0.8567^. 

3  log  0.8567  =  3  X  (9.9329  -  10)  =  9.7987  -  10 
9.7987  -  10  =  log  0.629.  Ans. 


63.   Find  by  logarithms  (f|)*. 

4  log  61     =4x1.7853  =7.1412 

4  colog  73  =  4  X  (8.1367  -  10)  =  2.5468  -  10 


9.6880  -  10  =  log  0.4876.  Ans. 


64.   Find  by  logarithms  (ff)». 

3  log     13  =  3x1.1139  =3.3417 

3  colog  71  =  3  x  (8.1487  -  10)  =  4.4461  -  10 


7.7878  -  10  =  log  0.006134.  Ans. 


65.   Find  by  logarithms  {^f. 

5  log     16  =  5x1.2041  =5.0205 

5  colog   9  =  5  X  (9.0458 -10)  =  5.2290- 10 


0.2495  =  log  17.76.  Ans. 

66.  Find  by  logarithms  {^f. 

3  log     35  =  3x1.5441  =4.6323 

3  colog   4  =  3  X  (9.3979  -  10)  =  8.1937  -  10 

2.8260  =  log  699.9.  Ans. 

67.  Find  by  logarithms  (5^)*. 

2  loj.^     60  =  2  X  1.7782  =  3.5564 

2  colog  1 1  -  2  X  (8.9586  - 10)  =  7  9172-10 

1.4736  =  log  29.76.  Ans. 


TEACHERS*    EDITION.  409 


68.    Find  by  logarithms  (43^)^. 

3  log      128  =  3  X  2.1072  =  6.3216 

3  colog    31  =  3  X  (8.5086  -  10)  -  5.5258  -  10 


1.8474  =  log  70.37.  Ans. 


69.   Find  by  logarithms  {2^f. 

5  log     103  =  5  X  2.0128  =  10.0640 

5  colog    37  =  5  X  (8.4318  -  10)  =    2.1590-10 


2.2230  =  log  167.1.  Ans. 

70.  Find  by  logarithms  (f|})». 

3  log     871  -  3  X  2.9400  =  8.8200 

3  colog  711  =  3  X  (7.1481  -  10)  =  1.4443  -  10 

0.2643  =  log  1.838.  Ans. 

71.  Find  by  logarithms  133. 

ilog    13=    i  of  1.1139  =  0.3713  =  log 2.351.  ^ns. 

72.  Find  by  logarithms  29^ 

I  log   29  =    I  of  1.4624  =  0.2925  =  log  1.961.  Ans. 

73.  Find  by  logarithms  879tV 

yV  log  879  =  j\  of  2.9440  =  0.2944  =  log  1.97.  Ans. 

74.  Find  by  logarithms  0.609?. 

log  0.609  =  9.7846  -  10 
30.  -  30 


4)39.7846  -  40 

9.9462  -  10  =  log  0.8834.  Ans. 

75.   Find  by  logarithms  93.73^ 

I  log  93.73  =  I  of  1.9719  =  0.9860  =  log  9.683.  Ans. 


410  ARITHMETIC. 


76.  Find  by  logarithms  21.97^ 

^  log  21.97  =  f  of  1.3418  =  1.1182  =  log  13.13.  Ans. 

77.  Find  by  logarithms  7.935^ 

f  log  7.935  =  f  of  0.8996  =  0.6426  =  log  4.391.  Ans. 

78.  Find  by  logarithms  0.8151. 

log  0.815  =.9.9112 -10 
3 


29.7336  -  30 
10  -10 


4)39.7336-40 

9.9334  -  10  =  log  0.8578.  Ans. 

79.  What  weight  of  sulphuric  acid,  specific  gravity  1.841,  will  fill 
a  silver  sphere  J 38"'"'  in  diameter? 

log     1383  _  6.4197 

log  0.52.'.6  =  9.7190 -10 

log   1.841-0.2650 

6.4037  =  log  2534000. 

That  is,  2534000<»""»  =  2.534»'«.  Ans. 

80.  What  is  the  area  of  a  circle  13.75  in.  in  diameter? 

log  6.875'*  =  1.6746 
log  3.1416  =0.4971 

2.1717  =  log  148.5. 

That  is,  148.5  sq.  in.  Am. 

81.  Find  the  depth  of  a  cubical  bin  that  holds  75  bu. 

log  75  =  1.8751 

log  2150.42  =  3.3325 

3)5.2076 
1.7359  =  log  54.44. 
Tliat  is,  54.44  in.  Ans. 


TEACHERS     EDITION. 


411 


82.   Find  the  diameter  of  a  24-lb.  shot,  specific  gravity  7.6. 


log 

24  = 

=    1.3802 

log 

1728  = 

=    3.2375 

colog 

7.6  = 

=    9.1192- 

-10 

colog 

62.5  = 

=    8.2041- 

-10 

colog  ( 

).5236  = 

=  10.2810- 

-10 

3)2.2220 

0.7407  = 

=  log 

log  5.504. 


That  is,  5.504  in.  Am. 


Exercise  LXXXIV. 


What  number  is  3  less  than  its  square  ? 

Assume  2.3  and  2.4. 

2.32  -  2.3  =  5.29  -  2.3  =  2.99,  an  error  of  -  0.01. 

2.42  -  2.4  =  5.76  -  2.4  =  3.36,  an  error  of  +  0.36. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.37. 
Hence,  error  of  2.3  :  0.1  :  :  0.01 :  0.37, 
or,  error  of  2.3  :  0.1  :  :  1 :  37. 

1x0.1 


Therefore,  the  error  of  2.3  = 
2.3-^0.0028  =  2.3028.  Ans. 


37 


0.0028. 


2.  A  flag-staff  50  ft.  high  broke,  and  the  top  falling  over  rested 
one  end  on  the  stump  and  the  other  17  ft.  from  its  base.  How  high 
was  the  stump  ? 

Assume  22  ft.  and  23  ft. 


412  ARITHMETIC. 


222  +  172  _  484  +  289  =  773,  an  error  of  -  11. 
23»  +  172  =  529  +  289  =  818,  an  error  of  +  89. 
The  difference  of  the  assumed  nnmbers  is  1,  and  the  difference 

of  the  errors  is  100, 
Hence,  the  error  of  22 :  1 :  :  11  :  100. 


Therefore,  the  error  of  22  =  12<J1  =  0.11. 
100 

22  +  0.111  =  22.11  ft.  Ans. 


3.   What  number  added  to  eight  times  its  reciprocal  is  equal  to  8  ? 
Assume  1.1  and  1.2. 

1.1  -^.  8  X  —  =  1.1  +  7.273  =  8.373,  an  error  of  +  0.373. 

1.2  +  8  X  —  =  1.2  +  6.667  =  7.867,  an  error  of  -  0.133. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.506. 
Hence,  the  error  of  1.1 :  0.1 :  :  0.373  :  0.506, 
or,  the  error  of  1.1  :  0.1 :  :  373  :  506. 

Therefore,  the  error  of  1.1  =  IAI^  =  0.0737. 
506 

1.1+0.0737=1.1737.  Ans. 


Assume  6.8  and  6.9. 

6.8  +  8  X  ;^  =  6.8  +  1.177  =  7.977,  an  error  of  -  0.023. 

6.8 

6.9  +  8  X  ~  =  6.9  +  1.159  =  8.059,  an  error  of  +  0.059. 

6.9 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.082. 
Hence,  the  error  of  6.8  :  0.1  : :  0.023  :  0.082, 
or,  the  error  of  6.8  :  0.1  :  :  23  :  82. 

Therefore,  the  error  of  6.8  =  ^-^7^  =  0.028. 
82 


6.8  +  0.028-6.828.  Atii, 


teachers'  edition.  413 


4.    Find  a  number  whose  reciprocal  is  equal  to  4  minus  the  number. 


Ai 

3sum 

e  3  and  4. 

i 

=  4- 

-3,  ori  = 

1. 

an  error 

of  +  f. 

i 

=  4- 

-  4,  or  i  = 

0, 

an  error 

of- 

-i 

The  difference  of  the  assumed  numbers  is  1,  and  the  difference 

of  the  errors  is  |-|. 
Hence,  the  error  of  3  :  1  :  :  | :  H, 
or,  the  error  of  3  :  1 :  :  8  :  11. 

Therefore,  the  error  of  3  =  ^-^  =  0.73. 

3  +  0.73  =  3.73.  Ans. 


Assume  0.26  and  0.27. 

—  =  4  -  0.26,  or  3.846  =  3.74,  an  error  of  -  0.106. 
0.26 

-^  =  4  -  0.27,  or  3.704  =  3.73,  an  error  of  +  0.026. 
0.27 

The  difference  of  the  assumed  numbers  is  0.01,  and  the  difference 

of  the  errors  is  0.132. 

Hence,  the  error  of  0.26 :  0.01 :  :  0.106  :  0.132, 

or,  the  error  of  0.26  :  0.01 : :  53  :  66. 

Therefore,  the  error  of  0.26  =  ^'^^^  ^^  =  0.008. 

66 

0.26  +  0.008  =  0.268.  Ans. 


5.   What  number  is  ten  times  its  own  logarithm  ? 

Assume  1.3  and  1.5. 

1.3  =  10  X  0.1139,  or  1.3  =  1.139,  an  error  of  -  0.161. 

1.5  =  10  X  0.1761,  or  1.5  =  1.761,  an  error  of  +  0.261. 

The  difference  of  the  assumed  numbers  is  0.2,  and  the  difference 

of  the  errors  is  0.422. 
Hence,  the  error  of  1.3  :  0.2  :  :  0.161 :  0.422, 
or,  the  error  of  1.3  :  0.2  :  :  161  :  422. 

Therefore,  the  error  of  1.3  =  '""'^o  "^  =  ^'^^^ 
422 


414  ARITHMETIC. 


Assume  1.37  and  1.38. 

1.37  =  10  X  0.13G7,  or  1.37  =  1.367,  an  error  of  -  0.003. 

1.38  =  10  X  0.1399,  or  1.38  =  1.399,  an  error  of  +  0.019. 

The  difference  of  the  assumed  numbers  is  0.01,  and  the  difference 

of  the  errors  is  0.022. 
Hence,  the  error  of  1.37  :  0.01  :  :  0.003  :  0.022. 
or,  the  error  of  1.37  :  0.01  :  :  3  :  '^•^ 

Therefore,  the  error  of  1.37  =  ^-^^^  -  0.0013. 

Therefore  the  number  is  1.37  +  0.0013  =  1.3713.  Ans. 
The  number  10  also  satisfies  the  conditions. 


6.   What  number  is  double  its  own  cube  root  ? 

Assume  2.8  and  2.9. 

2v^  =  2.82,  an  error  of  +  0.02. 

2v^  =  2.85,  an  error  of  -  0.05. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.07. 
Hence,  the  error  of  2.8  :  0.1  :  :  0.02  :  0.07, 
or,  the  error  of  2.8  :  0.1 :  :  2  :  7. 

Therefore,  the  error  of  2.8  =  ^^-^  =  0.0284. 

2.8  +  0.0284  =  2.8284.  Ans. 


7.   What  number  exceeds  its  cube  root  by  6^? 

Assume  8.2  and  8.3. 

8.2  -  y/s:2  =  6.184,  an  error  of  -  0.066. 

8.3  -  v^  =  6.275,  an  error  of  +  0.025. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.091. 
Hence,  the  error  of  8.2  :  0.1  :  :  0.066  :  0.091, 
or,  the  error  of  8.2 :  0.1 :  :  66  :  91. 

Therefore,  the  error  of  8.2  =  2iJp§  »  0.072. 

8.2  +  0.072  =  8.272.  Ayis. 


TEACHERS     EDITION.  415 

8.    What  is  the  number  which  added  to  its  own  square  makes  11  ? 

Assume  2.8  and  2.9. 

2.8  +  2.82  _  10.64,  an  error  of  -  0.36. 

2.9  +  2.92  =  11.31,  an  error  of  +  0.31. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.67. 
Hence,  the  error  of  2.8  :  0.1  :  :  0.36  :  0.67, 
or,  the  error  of  2.8  :  0.1 :  :  36  :  67. 

Therefore,  the  error  of  2.8  =  ^lA^  =  0.054. 

67 

2.8  +  0.054  =  2.854.  Ans. 


9.   What  is  the  number  which  multiplied  by  10  makes  8  more  than 
the  square  of  the  number  ? 

Assume  9.1  and  9.2. 

10  X  9.1  =  9.P  +  8,  or  91  =  90.81,  an  error  of  -  0.19. 

10  X  9.2  =  9.22  ^  8,  or  92  =  92.64,  an  error  of  +  0.64. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.83. 
Hence,  the  error  of  9.1 :  0.1 : :  0.19  :  0.83. 
or,  the  error  of  9.1  :  0.1 :  :  19  :  83. 

Therefore,  the  error  of  9.1  =  ^^^^  =  0.023. 

83 

9.1  +  0.023  =  9.123.  Ans. 


Assume  0.87  and  0.88. 

10  X  0.87  =  0.872  +  8,  or  8.7  =  8.7569,  an  error  of  +  0.0569. 
10  X  0.88  =  0.882  +  8,  or  8.8  =  8.7744,  an  error  of  -  0.0256. 
The  difference  of  the  assumed  numbers  is  0.01,  and  the  differ- 
ence of  the  errors  is  0.0825. 
Hence,  the  error  of  0.87  :  0.01  :  :  0.0569  :  0.0825, 
or,  the  error  of  0.87  :  0.01 :  :  569  :  825. 

Therefore,  the  error  of  0.87  =  ^'^^^^^^^  =  0.0069. 

825 

0.87  -H  0.0069  =  0.8769.  Ana. 


416  ARITHMETIC. 


10.    A  certain  number  is  equal  to  the  sum  of  ^  its  own  cube  plus 
^  its  own  square.     What  is  the  number  ? 

Assume  1.81  and  1.82. 

h^  +  Ml!  =  1.8073,  an  error  of  -  0.0027. 
6  4 

h^  +  M?!  ^  1.8329,  an  error  of  +  0.0128. 
6  4 

The  difference  of  the  assumed  numbers  is  0.01,  and  the  differ- 
ence of  the  errors  is  0.0156. 
Hence,  the  error  of  1.81 :  0.01 : :  0.0027  :  0.0156, 
or,  the  error  of  1.81 :  0.01 :  :  9  :  52. 

Therefore,  the  error  of  1.81  =  2:^l2<i  =  0.0017. 

52 

1.81  +  0.0017  =  1.8117.  Ans. 


11.   What  number  is  equal  to  its  square  minus  three  times  its 
logarithm  ? 

Assume  1.1  and  1.2. 

1.1»  -  3  X  0.0414  =  1.0858,  an  error  of  -  0.0142. 

1.22  _  3  X  0.0792  =  1.2024,  an  error  of  +  0.0024. 

The  difference  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.0166. 
Hence,  the  error  of  1.1 :  0.1 : :  0.0142  :  0.0166, 
or,  the  error  of  1.1 :  0.1  :  :  71  :  83. 

Therefore,  the  error  of  1.1  =  MilZl  =  0.086. 

83 

1.1  +  0.086  =  1.186.  A718. 


12.  The  sum  of  the  square  and  the  square  root  of  a  number,  being 
divided  by  1  plus  the  number,  gives  a  quotient  of  2^.  What  is  the 
number  ? 

Assume  2.7  and  2.8. 

2-7;  + ^^,  8^3  ,2.415,  an  emr  of -0.085. 
1  +  2.7  3.7 


teachers'  edition.  417 

MIW^S  ^  0513  ^  2.503,  an  error  of  +  0.003. 
1  +  2.8  3.8 

The  difFerence  of  the  assumed  numbers  is  0.1,  and  the  difference 

of  the  errors  is  0.088. 

Hence,  the  error  of  2.7  :  0.1 : :  0,085  :  0.088, 

or,  the  error  of  2.7  :  0.1  :  :  85  :  88. 

Therefore,  the  error  of  2.7  =  ^^^— ^  =  0.096. 

88 

2.7  4-  0.096  =  2.796.  Ans. 


Exercise  LXXXV. 

1.  Convert  ^,  jf ,  ^^j,  Vr  ^^^  continued  fractions. 
(1) 
3)11(3  ...A  =  -^^- 

2)3(1  ^^n     ,    1 


2 

1)2(2 
2 


'^1 


(2) 

13)75(5  .•.13  =  -^-^- 

10)13(1  ^"^1^1 

10  1+ — I 

3)10(8  3+- 

"1)3(3 
3 


418  ARITHMETIC. 


7)11(1  1  + i 

4)7(1  1+1 

4  a 

3)4(1 
3 

1)3(3 
3 

(4) 
W-2/^. 

1)7(7  ^  +  7 

7 


2.    Find  the  approximate  values  of  f ^ ;  f f ;  f|f . 

(1) 

20)27(1  .    ,o  =  _l_  1.1. 

20  ■■  '^     1+     1  1 


7)20(2  "  ^  1 


14 


2  +  -^  _l_-2 


6)7(1                                A +6                      1  + 
6                                         *" 
1)6(6  _1 =  3 

1 


6  1+14 


^^1 
1.  I  |.  ^n«. 
(2) 
H=1,V  1  =  1. 


^)f  .MA  =  i.-^  1.1  =  1 

1)5(5  7+1     « 

5  A 

1,  f,  f.  Ann. 


734)851(1 
734 


teachers'  edition.  419 


(3) 


117)734(6  ^  "^ \ 

702  6  +  -1 


32)117(3  3  +  -^ — - 


_96  1  + 

21)32(1  1  +      ^ 


21  1  +  1 

11)21(1  10 

n 

10)11(1 
10 

1)10(10 
1      1  10  1  44 


1  -.   .        1  51 

1         6 


1  + 


1  19  ^1 


,         1        22  1  _69 

6  +  1  i  +  -L_ 

"*  6+       ' 


25  3+      1 


1  +  ^        29  1^^ 

^■^1  1,  f,  H,  M,  «,  M.  4,:.. 

3.   Find  common  fractions  approximating  to  0.236 ;  0.2361 ;  1.609. 

(1) 
0.236  =  ^3^%=//^. 

59)250(4  .     5,  1 

236  ••  ^^TF  . 

—  4  + 


14)59(4                                --             1 
56  4  + 

3)14(4  4  +  - 


1 

1 


12  1+2 

2)3(1  "^ 

2 

1)2(2 
2 


420 

ARITHMETIC. 

1       ! 

1                     21 

4     4 

A.       I             89 

1          4 

-1  " 

-H 

1            17 

h  A.  H.  li  ^^- 

4^.     1        72 
*4 

(2) 

0.2361 . 

-  JUAJL 

2361)10000(4 
9444 

.-. 

fWs 

1 
'     .        1 

556)2361(4 
2224 
137)556(4 
548 

8)137(17 
136 
1)8(8 
8 

1.  1 

-"-1 

1     1 

1                     293 

4     4 

1  1       1               ^241 

1         4 

•-^ 

1           17 

i  iV.  H.  A'^-  ^^- 

4+     1       T2 

4  + 


teachers'  edition.  421 


(3) 
1.609  =  1+T%V^. 
609)1000(1  ,.  i^^eo.^^l+_I 


609  ••    "^TT^XFT^--^  -^ 


391)609(1                                   "  ^  ^           1 
391  1  + 


218)391(1  1  + :; 

218  1+— 1— - 

173)218(1  3  +  — i- 

173  l  +  _ 


1 


45)173(3  5  + 


1_ 
'135'  2  +  - 

38)45(1  3 

38 
7)38(5 
35 
3)7(2 
6 

1)3(3 
3 


13  1  ^ 


^4  !.-> 


1  8  '^:—i 


1+    '   ,      =-  .         1+ 


1+^  2,  f,  f,  f,  ft,  \i.  Am. 


422 


ARITHMETIC. 


4.   Find  common  fractions  approximating  to  0.382 ;  1.732 ;  0.6253. 

(1) 

0.382  =  ,3^V^  =  i^. 

1 


191)500(2 

382 


-'-m 


118)191(1 
118 

73)118(1 
73 


1  + 


1  + 


1  + 


45)73(1 
45 


1  + 


28)45(1 

28 


1  + 


1  + 


17)28(1 
17 


1  + 


1  + 


11)17(1 
11 


6)11(1 
_6 

5)6(1 
5 


1)5(5 
5 


2  + 


1_1 

2     2 

1__1 
3' 


2  + 


1  + 


2  + 


1  + 


2  + 


1  + 


1  + 


1  + 


1  + 


2  + 


1  + 


1  + 


1  + 


1  + 


TEAOHEES'   EDITION.  423 


1  ^13  1  ^21 

1+       ^      .  1+       ^ 


i  +  -3-^  1+     ' 


1  +  ^^  1+     1 


1  +  -!-  1+     1 


1  _34 

1  +  -   ^ 


1  +  ^ 


l  +  -i 


1  +  -^ 


IH-^ 


■^-^i 


i.  h  h  i  A.  A.  M.  fi.  ft-  ■^^«- 

(2) 
1.732  =H|f 


l^^)???^  .-.  UM  =  1+-^ 


183  ••™      -^_^       1 


)183(2                                    ^  ■              1 
134  2  + 


_  1  1 

49)67(1  1  + T 

49  2  +  —^ 


18)49(2  1  +  — ^ — r 

36  2+—^ 


13)18(1  1+-^ 

13  1+1 

5)13(2  2 

3)5(1 
3 

2)3(1 

2 

1^2(2 
2 


424  ARITHMETIC. 


,,.,         .._^_^     .n. 


2  +  -L 


1  + 


J_-5.  1  + 


1 


2+i  2+      1 


1  1  +  -L 


2+       1 


1  +  ^—       =12.  1  +  -!-, 

2+       ' 


l+-i-.  ^26.  ,^       1 


l+-i-.  1^  2+      1 


2  +  -^  1+      ' 


l.-A.,  2.^ 

2.1  1.1 

2.  i  J.  H.  If.  H.  M.  ¥/-  ^ns. 

0.6253  =  ^<^^j. 

6253)10000(1  .     e253  1 

6253  ••■iirtnny=  j- 

3747)6253(1  ^  +  1 

3747  1  +  - 


2506)3747(1  1  +  — ; 

2506  2  + 


7)17(2  1241)2506(2  51+     ^ 


14  2482  1  I      1 

3)7(2  24)1241(51  o  ,  1 

6  1224  3 

1)3(3  17)24(1 

3  17 

7 


teachers'  edition.  425 


1      ^  1  ^262 


1+ — - 


1      2 


1+       ' 


1    ,  J-      ^  1 

1+T  24-  i 


1 

1 


'+  1 

51 +i 


1  +  i  1  781 


1+— t- 


1249 


5  1  +  ^ 


8  1+.     1 


^-^^-T  ^2.      1 


2 

257 


1  +  _i 411  1.  i  f.  f .  ffi.  Hi  AV  -471S. 


1  + 


51 


5.   Find  approximate  values  of  ^  ;  f^f  ;  |^^  ;  f  ||. 

(1) 

171)457(2  .    1.1  1 

342  "  ^^^  ~  o  1 

2  + 


115)171(1  I4._i 

115  1+ 1 

56)115(2  2  + - 

112  18+—^ 


3)56(18  1  +  ^ 

54  ^ 

T  1  =  1.  -3-  =  l 

1)2(2  2     2  2  +  i   ^ 

2  1 


426  ARITHMETIC. 


1^-i  !.-•- 


2  +  _L 


18  +  1 


55  1 


2  +  -^    147 

1  +  — L_  i  i.  I,  T^^.  i^A-  ^^• 

2  +  — 
18 


(2) 
613)757(1  .  «ig    1 


5_76 
37)144(3 


33 


1         13 


.-.  m- 


613  ••-   ^^   1 


144)613(4  "  ^  ^     1 


3  +  -A 


m  1  +  -^ 

33)37(1  8  +  ^ 


4)33(8 
32  l_j       1  ^4 

1)4(4  1   •     1+1  ^' 

^  4 


_J ^149 

3  3+—!— 

17  '4 


1  +  _L    21 

3  +  1 


teachers'  edition.  427 


(3) 


1t¥  =  f  f  t- 


237)271(1 
237 


34)237(6 
204 


33)34(1 
33 


1)33(33 
33 


5)14(2 
10 


-■-m 

1  ,        1 

6,       1 

i_i 

1 

1             7 

1    • 

1         6 

1,  f,  |.  Am. 

(4) 


W  =  8A.  .,8AV  =  8+^ 


3+       1 


33)113(3  9  ^       1 
^  2+     1 

14)33(2  rri 

28  1+5 


8  =  8. 


4)5(1  .         8+i=8i. 

^)f  8+^  =  8f. 


8  +  -^     =8tV  8+-J— -  =8^V 

3+-^  3+       ^ 


2+1  2+     1 


2  2  +  1 


8,  8i,  8f,  8tV,  8/j.  ^m. 


428  ARITHMETIC. 


6.    Find  the  proper  fraction  that,  when  reduced  to  a  continued 

fraction,  will  have  2,  3,  5,  6,  7  as  quotients. 

1                          V09      ^                      1_7. 

1        43  . 

o  .        1                   1640    """"                 6|     43 ' 

5^     222 ' 

'^    1           ^                                                                    1             999 

1         709 
2m     1640 

5  +  ^                                   3^     709' 
6  +  1 

7.   Find  a  series  of  fractions  approximating  to  the  ratio  of  the 
pound  troy  (5760  grs.)  to  the  pound  avoirdupois  (7000  grs.). 


nn=m-  .  i44  =  __i 


i^= 


144)175(1  1  +  — - 

144  4  + 


31)144(4  1  + L 


124  1+,      1 


20)31(1  ._l 


20 


1  + 


11)20(1  ^  +  2 

n 

9)11(1  1  =  1  ^         =5 

-^  1     '       1  +  ^-    ' 

2W  ,         ,  4^1 


1)2(2  1  +  1     5 

2  4 


9 


1  11 


^^        I  4+       ^ 


^^I  1^^ 


-^1 


1  +  _1  17  4 

4+       1 


l  +  _i_  1.  t.  i  A.  if  H.  ^'w- 

1 

i 


i+i 


teachers'  edition.  429 


8.   Find  a  series  of  fractions  approximating  to  the  ratio  of  the  side 
of  a  square  to  its  diagonal ;  that  ratio  being  1  :  1.414214  nearly. 

1  1000000      7071 


1.414214      1414214      10000 


7071)10000(1  .     ,_ojLi_  _       1 

7071  ■■  Toooo  -| 

2929)7071(2  ^  +  1 

5858  2  + 

1213)2929(2  2  +  — 


2426  2  +  — 

503)1213(2  2  -f       ^ 


1 

=  1. 

1 

2 
3" 

1 

5 

--'i 

7 

1 

12 

1 

2  +  ^ 

17 

1006  2  I        ^ 

207)503(2  3  ,  _L_ 

89)207(2  2 

178 

29)89(3 
87 
2)29(14 
28 
1)2(2 
2 


1  _29 

2+       ' 


2  +  -A 


2  +  1 


1  ^70 

2  +  -  1 


2  +  -^ 


2  +  ^ 


-I 


1.  f ,  i  {h  ih  U-  ^ns. 


430  ARITHMETIC. 


9.   Find  a  series  of  fractions  approximating  to  the  ratio  of  the  ar 
to  the  square  chain,  from  the  equality  1  ar  =  0.2471  of  a  square  chain. 

0.2471  =  AW5. 
2471)10000(4  _..^.^ ^ 


0884 


4  + 


116)2471(21 

2436  21  + 


35)116(3 
105 


3  + 


11)35(3  5  +  1 

33  2 

2)11(5 
10 
1)2(2 

2 

1  =  1  1     _2i  1         ^eA 

4     4  4  .  JL     85  4  +  -^_      259 

214 

10.   Find  a  series  of  fractions  approximating  to  the  ratio  of  the  48- 
pound  shot  to  the  weight  of  the  French  shot  of  2A^«. 
48  lbs.  =  48  X  0.45359  =  21.77232^«. 

907)1000(1  21.77232  ^  907 

907  24  1000 


^^m^  •••  W.=-^ 


837  '"""      j_^       1 


70)93(1  9^_1 

1  + 


23)70(3  \^1 

1)23(23 
23 

1     1  1     _  9  1  _10  1  ^39 

i'   '        1+1     !«■  1+-1-     11'  1  +  -1-    ~'' 

^  9  +  1  9  +  ^— 

1  1+r 


1.  A.  H.  if.  ^n,. 


3 


teachers'  edition.  431 

11.  If  the  mean  diameter  of  the  Earth  is  reckoned  at  7912  mi., 
and  that  of  Mars  4189  mi.,  find  a  series  of  fractions  approximating 
to  the  ratio  of  the  mean  diameters  of  these  two  planets. 

4189)7912(1  .    4189  __J 

4189  ••  7^ 


3723)4189(1  ^  +  i 


3723  1+         . 

466)3723(7  7  +  - 

3262  ^ 

461)466(1 
461 


1=1.  -^—         =^. 

1,1      2 


^'  T5'  XT' 


1  +  -^    15 
^4 


12.    Find  a  series  of  fractions  approximating  to  the  ratio  of  a  cubic 
yard  to  a  cubic  meter  from  the  equality 

1  cu.  yd.  =  0.76453  of  a  cubic  meter. 

0.76453  =  tWAV 

76453)100000(1  .,^5  3     1 

76453  •  •  T^^^os 

23547)76453(3 
70641 


5812)23547(4 

23248 


299)5812(19 
5681 
131)299(2 
262 

~d7 


432  ARITHMETIC. 


1  =  1  1  250 

1        '  ^   ^         t  327 

J_  =  3  3  +  — L- 

1+1     ^  4  +  -1- 

^+3  19 


1  + 


-t 


13.   Find  a  series  of  fractions  approximating  to  the  ratio  of  the 

kilometer  to  the  mile,  from  the  equality    1"  =  1.09362  yds. 

!"»  =  1.09362  yds.  P™  =  1093.62  yds.  l"""  =  0.621  mi. 

0.621  =Tm- 
621)1000(1  .     g2i  _       1 

621  •  •  TSVff  -  1 

379)621(1  ^  "^  ~ 


379 
242)379(1 


1  + 


^^^-T 


242  1  + 

137)242(1  1  +  1 

137  ^ 

105)137(1 
105 

32)105(3 
__96 
9)32(3 
27 
5 

1  =  1  1  3 

^     1.  ^^rtl 


1  +  12 


1 


1  + 


2  1+       ' 


1+1  1+1 


TEACHERS     EDITION. 


433 


1  + 


1  + 


18 
29' 


1  + 


1  + 


1  + 


1  + 


1  + 


59 
95* 


1  + 


-I 


Exercise   LXXXVI. 


1.  Find  the  seventh  term  of 
the  series  3,  5,  7 

3  +  (6  X  2)  =  3  +  12  =  15.  Ans. 

2.  Find  the  fifteenth  term  of 
the  series  2,  7,  12 

2  +  (14  X  5)  =  2  +  70  =  72.  Ans. 


3.  Find  the  sixth  term  of  the 
series  2,  2f,  3f 

2  +  (5xf)  =  2  +  3f  =  5f  Ans 

4.  Find  the  twentieth  term  of 
the  series  2,  3^,  4| 

2  +  (H  X  19)  =  2  +  23f  =  25|. 


5.   Find  the  seventh   term   of 
the  series  21,  19,  17 

21 -(6x2)  =  21 -12  =  9.  Ans. 


6.    Find   the   twelfth  term  of 
the  series  18,  17^,  16f 

18  -  (11  X  I)  =  18  -  7i  =  10|. 


7.  When  the  first  terra  of  a 
series  is  5,  and  the  common  dif- 
ference 2^,  find  the  thirteenth 
and  eighteenth  terms. 

5  +  (12x2i)  =  5  +  27   =32.    (1) 
5  +  (17x2i)  =  5+38i  =  43^.(2) 


8.  Find  the  common  difference 
in  a  series  whose  fourth  term  is 
12  and  seventh  term  27. 


27-12 


=  5.  Ans. 


9.  Find  the  common  difference 
in  a  series  whose  first  term  is  20 
and  fourth  term  40. 


10.  Find  the  common  differ- 
ence in  a  series  whose  first  term 
is  2  and  eleventh  term  20. 


20-2 
10 


Ans. 


434 


ARITHMETIC. 


11.  Find  the  common  differ- 
ence in  a  series  whose  third  term 
is  7  and  eighth  term  12^. 


121:^^  =  1.1. 

5 


Arts. 


12.  Find  the  common  differ- 
ence in  a  series  whose  first  term 
is  1  and  fourth  term  19. 


19-1 


6.  Ans. 


Exercise   LXXXVII. 


1.    Find  the  sum  of  14-5  +  9 
+ 10  twenty  terms. 


Z=l+(19x4)  =  77 

1  +  77 


«  =  20x 


=  780.  Ans. 


2.   Find  the  sum  of  4  +  5^  +  7 
+ to  eight  terms. 

Z  =  4  +  (7xH)  =  14i 
«»8xi^tiii=.74.  Ans. 


3.  Find  the  sum  of  8  +  7f  +  7^ 
+ to  sixteen  terms. 

Z=8-(15xi)  =  3. 

«  =  16  X  ^4^    ==  88.  Ans. 


4.  Find  the  sum  of  20  +  18^ 
+  16^  + to  seven  terms. 

Z  =  20-(6xl|)  =  9i 

,^7x20±9i    =miAm. 

5.  Find  the  sum  of  the  first 
twenty  natural  numbers. 

1  +  20 


«-20x 


210.  Am. 


6.   Find  the  sum  of  the  natural 
numbers  from  37  to  53  inclusive. 


8=17x 


37  +  53 


765.    Ans. 


7.  Find  the  sum  of  a  series  of 
thirty  terms,  of  which  the  first  is 
21  and  the  last  59. 


«  =  30x 


21  +  59 


1200.  A71S. 


8.  Find  the  sum  of  the  series 
whose  first  two  terms  are  3  and  9 
and  last  75. 

l  =  a  +  {n—l)d. 
75  =  3 +  6n- 6. 
6n=.78. 
n  =  13. 

,  =  13x^4^  =  507.  Ans. 


9.  Find  the  sum  of  a  series  of 
twenty  terms  whose  third  and 
fifth  terms  are  10  and  15. 


10  +  15 


-2i 


Z=-5  +  (19x2J)  =  52J. 
,=,20x^^^tM-575. 


Am. 


TEACHERS     EDITION. 


435 


10.  A  stone,  when  dropped  from  a  height,  falls  through  16.1  ft.  in 
the  first  second,  48.3  in  the  next,  80.5  in  the  third,  and  so  on,  in 
arithmetical  progression.  How  far  will  it  fall  in  the  seventh  second? 
and  how  far  in  7  sec.  ? 

7th  term  is  16.1  +  (6  X  32.2)  =  209.3  ft.  (1)  Ans. 


7x 


16.1  +  209.3 


■■  788.9  ft.  (2)  Ans. 


11.  A,  who  travels  8  mi.  the  first  day,  11  the  second,  14  the  third, 
and  so  on,  overtakes  in  17  dys.  B,  who  started  at  the  same  time,  and 
travelled  uniformly.     What  is  B's  rate  per  day  ? 

Z  =  8  +  (16  X  3)  =  56. 

■  4-  56 


s  =  17x 


544. 


-^V  =  32mi.  Ans. 

12.  On^  hundred  stones  lie  in  a  straight  line,  1  yd.  apart.  A  boy 
starts  at  the  first  stone,  brings  each  of  the  others  in  separately,  and 
piles  them  with  the  first  stone.     How  far  does  he  travel  ? 

Z  =  2  +  (98  X  2)  =  198  yds. 
8=  99  X  1(2  +  198)  -  9900  yds. 

9900  yds  =  5f  mi.  Ans. 


Exercise  LXXXVIII. 


1.    Find  the  eighth  term  of  the 
series  2,  6,  18 

2  X  3^  =  2  X  2187  =  4374.  Ans. 


2.   Find  the  fifth  term  of  the 
series  8,  4,  2 

8  X  ihf  ^  8  X  tV  =  h  ^^s- 


3.    Find  the  seventh   term   of 


2  X  (f )«  =  2  X  -W-  =  22f  |.  Ans. 

4.    Find  the  sixth  term  of  the 
series  4,  2f,  1^ 

4x(f)^  =  4x^  =  H«-  ^ris. 


436 


ARITHMETIC. 


5.    Find  the  eighth  term  of  the 
series  4,  10,  25 

4x(fy  =  4xi||F  =  244m. 


6.  Write  the  first  three  terms 
of  the  series  whose  fifth  and  sixth 
terms  are  112  and  224,  respec- 
tively. 


7.  (1)  Ans. 


r  =  2. 
112 

2* 

2x7  =  14.  (2)  Ans. 
22  X  7  =  28.  (3)  Ans. 

7.  The  seventh  and  ninth 
terms  of  a  series  are  100  and  144, 
respectively.  Find  the  twelfth 
term. 


The  9th  term  =  7th  term  X  r*. 
•••  r^  =  W 

12th  term  =  144  x  (f)=» 
=  144  X  iU 
=  248.832.  Avs 

8.  A  capital  of  $1000  is  in- 
creased by  Y^^  of  itself  each  year. 
What  will  it  be  at  the  beginning 
of  the  fifth  year  ? 

f  1000  x(H)*  =  ?  1000  xiM^i 
=  $1464.10.  Am. 

9.  A  capital  of  $1000  is  in- 
creased by  j^jj  of  itself  each  yea". 
What  will  it  be  at  the  beginning 
of  the  sixth  year  ? 

$1000  X  (!§§)» 

=  $ioooxHMM5HS 

=  $1338.23.  Am. 


Exercise  LXXXIX. 

1.   Find  the  sum  of  2  +  6  -|- 18  -h to  six  terms. 

3«- 


2x 


3- 


2.   Find  the  sum 
«=«  1  x^^ 


3.   Find  the  sum 
3»- 


8  =  3x 


2x^F  =  728.  Am. 

of  1  -I-  2  -f  4  -F to  nine  terms. 

=  lX^P  =  511.  Am. 


of  3  -f  9  -f  27  + to  five  terms. 

==3x^F  =  363.  An.<;. 


4.   Find  the  sum  of  2  -h  3  -I-  4 J  -I- to  eight  terms. 

a  »  2  X  ^D-::i  =  2  X  ^^^  =  98|?.  Am. 
i-1  i 


teachers'  edition.  437 

5.  Find  the  sum  of  1  +  I  +  ^  + to  eight  terms. 

s^lX  V^  =  ^  ~  r''^  =  If  f  f  X  I  =  IMf  f  Ans. 

6.  Find  the  sum  of  1  +  |^  +  |  + to  ten  terms. 

1  —  ^i^lo  1  _    _!_ 

J-  —  ^  ^- 

7.  Find  the  sum  of  |  +  I  +  f  + to  eight  terms. 

-•■  -  3  J 

8.  Find  the  sum  of  the  first  six  terms  of  the  series  whose  first  term 
is  3  and  ratio  5. 

s  =  3  X  ^^^  =  3  X  i^P^  =  11718.  Ans. 
5-1  * 

9.  Find  the  sum  of  the  first  eight  terms  of  the  series  whose  first 
term  is  3  and  ratio  I. 

s  =  3x^^^'  =  3x^^^f^  =  3xeMx|  =  4fff.  Ans. 

10.  A  person  saved  in  one  year  $64,  and  in  each  succeeding  year, 
for  9  years  more,  1^  times  as  much  as  in  the  preceding  year.  Find 
the  whole  amount  saved. 

(  3  \10  _  1  ,F;fi  6  8  1 

s  =  $  64  X  ^-^ =  1 64  X  ^^^-2-^  =  $  7253.13.  Ajis. 

f  —  1  i 


Exercise  XC. 


1.    Find  the  sum  of  the  infi- 


1  1 

_-^_  =  2  =  1.  Ans. 

1-i     i 


2.   Find  the  sum  of  the  infi- 
nite series  1  +  1+27  + 

s  =  -^  =2=2.  Ans. 


4d8                                             ARITHMETIC. 

3.   Find  the  sum  of  the  infi- 

6.   Find  the  sum  of 

the  infi- 

nite  series  j  +  i*^  +  ^j  + 

nite  series  0.212121 

-r^rh^-^"'- 

^         0.21         0.21 
1  -  0.01      0.99 

•■>l 

—.Am. 

99 

4.   Find  the  sum  of  the  infi- 

7.  Find  the  sum  of 

the   iufi- 

nite  series  i  +  ^V  +  jhz  + 

nite  series  0.9999 

'=r^i=f=i-^'"- 

s_     0.9         0.9      J 
1  -  0.1     0.9 

.  Ans. 

5.   Find  the  sum  of  the  infi- 

8.   Find  the  sura  of 

the  infi- 

nite  series  0.171717. 

nite  series  0.232323 

s        017     _  0.17  _  17^^^ 
1-0.01      0.99     99 

0.23        0.23  _ 
1  -  0.01      0.99 

=  1- 

9.    Find  the  sum  of  the  infinite 

series  0.36848484 

0.0084       0.0084       84 
*     1-0.01       0.99       9900" 

36        84       3648 
100     9900     9900' 

Ans. 

10.   Find  the  sum  of  the  infinite 

series  0.15272727. 

^       0.0027    _  0.0027  _    27 
1  -  0.01       0.99       9900 

15        27   ^  1512 
100     9900     9900' 

Ans. 

Exercise  XCL 

1.   A  deposits  $60  in  a  savings  bank,  and  draws  it  out  at  the  end 
of  8  yrs.,  with  4%  compound  interest.     What  does  he  receive? 

log  ^  =  log  P  -I-  n  X  log  (1  -f-  r), 
log  ^  =  log  60  h  8  X  log  1.04. 

log  60  =  1.7782 
8  X  log  1.04  =  0.1360 


That  is,  $82.08.  Ans. 


1.9142  =  log  82.08 


2.  What  will  $  100  amount  to  in  7  years,  interest  at  8  %  per  annum, 
compounded  semi-annually  ? 

log  il  =  log  P  -I-  n  X  log  (1  +  r), 
log  il  =  log  100  +  14  X  log  1 .04. 


teachers'  edition.  439 

log  100  =  2.0000 
14  X  log  1.04  =  0.2380  =  log  173. 

2.2380 
That  is,  1 173.  Ans. 

3.  In  how  many  years  will  a  sum  of  money  double  itself  at  6%, 
compounded  annually  ? 

log  ^  =  log  P  +  n  X  log  (1  +  r), 
log  2  =logl  +nx  log  1.06, 
log  2  -logl=  nx  log  1.06. 

log2  -logl^^ 
log  1.06  ■ 

,.  0.3010  ^,^g^^_ 
0.0253 
That  is,  11.8  yrs.  Ans. 

4.  In  how  many  years  will  a  sum  of  money  treble  itself  at  6%, 
compounded  annually  ? 

log  J.  =  log  P  +  n  X  log  (1  +  r), 
log 3  =  log  1  +nx  log  1.06, 
log  3  -  log  1  =  n  X  log  1.06, 
log3-logl^^ 
log  1.06 

...0^771^  18.9  =  n. 
0.0253 

That  is,  18.9  yrs.  Ans. 

5.  In  how  many  years  will  $  87  amount  to  f  99  at  3  %,  compounded 
annually  ? 

log  .A  =  log  P  +  n  X  log  (1  +  r), 

log  99  =  log  87  +  n  X  log  1.03, 

log  99  -  log  87  =  n  X  log  1.03, 

log 99 -log 87  _^ 

log  1.03 

0.0561      .  oo     „ 

.•.  =  4.00  ^  n, 

0.0128 

That  is,  4.38  yrs.  Aiis. 


440  ARITHMETIC. 


6.  In  how  many  years  will  $100  amount  to  $175  at  4%,  com- 
pounded annually  ? 

log    A  =  logP+nxlog(l -l-r), 
log  175  =  log  100  +  w  X  log  1.04, 
log  175  -  log  100  =  n  X  log  1.04, 
log  175 -log  100     ^ 
log  1.04 

.•.^^i^=  14.29  =  n. 
0.00170 

That  is,  14.29  yrs.  Ans. 

7.  At  what  rate  per  cent  will  a  sum  of  money  double  itself  in  12 
years,  compound  interest? 

log  ^  =  log  P  +  n  X  log  (1  +  r), 
log  2  =  logl  +  12xlog(l+r), 
log  2  -  log  1  =  12  X  log  (1  +  r), 

l^&l^i2^  =  log(l+.). 

.-.  ^^^  =  0.0251  =  log  (1  +  r). 

.-.  1.0595  =  1  +  r,  or  r  =  0.0595. 
That  is,  5.95%.  Ans. 

8.  At  what  rate  will  a  sum  of  money  treble  itself  in  15  years,  at 
compound  interest  ? 

logA  =  logP+wxlog(l  +r), 
log  3  =  log  1  +  15  X  log  (1  +  r), 
log 3 -log  l  =  15x]og(l+r), 

i^S^^^  =  log(H-r). 

.-.  M221  =  0.0318  =  log(l  +  r). 
15 

.-.  1  +  r  =  1.076,  or  r  =  0.076. 

That  is,  7.6%.  Ans. 

9.  At  what  rate  will  $80  at  compound  interest  amount  to  $110  in 
8  yrs.  ? 

log    ^  =  log  P  +  n  X  log  (1  +  r), 
log  110  =  log  80  +  8  X  log  (1  +  r), 
log  110 -log   80-8xlog(l+r), 


teachers'  edition.  441 


log  110 -log  80      1      /I    ,     N 
-^ ^ ^—  =  log  (1  +  ^)- 

.-.  2iM83  =  0.0173  =  log(l+r). 
8 

.-.     l+r=  1.041,    or   r  =  0.041. 

That  is,  4.1%.  Ans. 

10.  What  sum  must  be  invested  at  5%,  compound  interest,  to 
amount  to  $1200  in  7  yrs. 

log       ^  =  logP+nXlog(l +r), 
log  1200  =  log  P  +  7  X  log  1.05, 
log       P=logl200-7xlogl.05. 

log  1200  =  3.0792 
7xlog  1.05  =  0.1484 

2.9308 

=  log  852.8. 
That  is,  $852.80.  Ans. 

11.  What  sum  must  be  invested  at  4%,   compound  interest,  to 
amount  to  $  2000  in  10  yrs.  ?     To  amount  to  $  5000  in  8  yrs.  ? 

log       ^  =  logP+nXlog(l +r), 
log  2000  =  log  P  +  10  X  log  1 .04, 
log       P=  log  2000- 10  X  log  1.04. 
log  2000  =  3.3010 
10  X  log  1.04  =  0.1700 

3.1310 
=  log  1352. 
That  is,  $1352.  (1)  Ans. 

log      ^  =  logP+nXlog(l +r), 
log  5000  =  log  P+  8  X  log  1.04, 
log       P  =  log  5000  -  8  X  log  1 .04. 
log  5000  =  3.6990 
8  X  log  1.04  =  0.1360 

3.5630 
=  log  3656. 
That  is,  $3656.  (2)  Am. 


442  ARITHMETIC. 


12.  If  A  puts  $100  a  year  into  a  savings  bank  that  pays  4%  per 
annum,  compound  interest,  what  will  he  have  in  the  bank  at  the  end 
of  10  years  ? 

From  ?  438, 

^a(r»-l)  _  1 100  X  0.479 

*^     r-1    *  *  0.04 

^^  $100(1.04^° -1)  ^_$47.9 

1.04-1       "  *      0.04' 

^^$100(1.479-1)  s  =  11197.50.  Ans. 

0.04 

13.  What  will  be  the  amount  in  the  last  problem  if  the  bank  pays 
4t%  per  annum? 

_  a(r»-l)  _  $  100(1  ■045^° -1)  _  $100(1.5525-1) 
r-1  1.045-1  0.045 

^  $100x05525  ^|55^^.j227  78    Am 
0.045  0.045 

14.  What  should  be  paid  to-day  for  an  annuity  of  $500  a  year, 
for  12  years,  if  money  is  worth  3^%,  compound  interest? 

_  a(r»-l)  _  500(1.035^^-1) 
*         r-1  0.035 

Present  worth  of  s  is  12  log  1.035  =  0.1788 

500  X  0.509  -  log  1.509. 

0.035  X  1.035"* 
log    500  =  2.6990 
log  0.509  =  9.7067 -10 
colog  0.035  =  1.4559 
colog  1.035  =  9.8212 -10 

3.6828 
-log  4818. 
That  is,  $4818.  Ans. 

15.  What  should  be  paid  to-day  for  an  annuity  of  $300  a  year, 
for  10  years,  if  money  is  worth  4%,  compound  interest? 

300(1.04^0-1) 
* "  0.04 


teachers'  edition.  443 

Present  worth  of  s  log  1.04  =  0.0170 

300  X  0.479  ^ 

0.04xl.04i«'  0.1700 

log    300  =  2.4771  =  log  1.479. 

log  0.479  =  9.6803  -  10 
colog0.04    =1.3979 
colog  1.04"'  =  9.8300  -  10 

3.3853 

=  log  2428. 
That  is,  $2428.  Ans. 

16.  What  should  be  paid  to-day  for  the  assurance  that  5  years 
hence  I  shall  begin  to  receive  $500  a  year,  for  8  years,  if  money  is 
worth  4|,  compound  interest? 

^_;  500(1.0458-1) 
0.045 

Present  worth  of  s  is  log  1.045  =  0.0191 

500  X  0.422  ^ 

0.045  X  1.04513'  0.1528 

log      500  =  2.6990  =  log  1.422. 

log  0.422  =  9.6253-10 
colog   0.045  =  1.3468 
colog  1.045^3  =  9.7517 -10 

3.4228 
=  log  2647.50. 
That  18,  $2647.50.  Ans. 

17.  If  interest  is  reckoned  at  6  %,  what  sum  of  money  must  be  paid 
.nually,  beginning  a  year  hence,  to  clear  off  a  debt  of  $10,000  in 

5  equal  payments  ? 

s  =     "       ~     ,  the  amount  of  $1  annually  deposited  at  6%  for  5  yrs. 
0.06 

Each  payment  must  be   ^0^00^^-^^'  ^  10000  x  1.06^  x  0.06^ 
^  ^  (1.06^-1)  0.338 

0.06 


ann 


444 


ARITHMETIC. 


log  10000  =  4.0000 
log  1.06*  =  0.1265 
log  0.06  =8.7782-10 
colog  0.338  =  0.4711 

3.3758  =  log  2376. 


log  1.06  =  0.0253 
5 


0.1265  =  log  1.338. 
That  is,  $2376.  Ans. 


18.  If  interest  is  reckoned  at  6%,  what  is  the  amount  of  each  of 
12  equal  semi-annual  payments,  the  first  to  be  paid  6  months  hence, 
required  to  clear  off  a  debt  of  1 24,000  ? 

CI  03^^  —  1) 
s  =  ^  '         — \  the  amount  of  $1  annually  deposited  at  3%  for 
yj.yJo 

12  yrs. 
Each  payment  must  be  ^^QQ^X^^^"  =  24000  x  1.03^' X  0  03. 
^  ^  (1.03^"-!)  0.424 


0.03 

log  24000  =  4.3802 
log  1.03^2  _  0.1536 
log    0.03  =  8.4771-10 
colog  0.424  =  0.3726 

3.3835  =  log  2418. 


log  1.03  =  0.0128 
12 

0.1536  =  log  1.424. 
That  is,  12418.  Am. 


Miscellaneous  Problems. 


1.   Make  six  different  numbers 

by  logarithms,   their  co 

with  the  digits  1,  2,  3,  and  find 

product. 

their  sum. 

235  X  253  X  325 

123 

X  352  X  523  X  532. 

132 

log  235  =  2.3711 

213 

log  253  =  2.0431 

231 

log  325  =2.5119 

312 

log  352  =  2.5465 

321 

log  523  =  2.7185 

1332.  Ans, 

log532=.  2.7259 

2.   Make  six  different  numbers 
with  the  digits  2,  3,  5,  and  find, 


continued 


15.2770 
=  log  1,892,000,000,000,000. 


TEACHERS     EDITION. 


445 


3,  Make  six  different  numbers 
with  the  digits  8,  7,  3,  and  find, 
by  logarithms,  their  continued 
product. 

873  X  837  X  783  . 
X  738x387x378. 
log  873  =  2.9410 
log  837  =  2.9227 
log  783  =  2.8938 
log  738  =  2.8681 
log  387  =  2.5877 
log  378  =  2.5775 

16.7908 
=  log  61,770,000,000,000,000. 

4.  Find,  by  logarithms,  the 
missing  term  in  each  of  the  fol- 
lowing proportions : 

(1) 
7.13  :  3.57  :  :  4.18  :  ? 

3.57  X  4.18  ^  ^ 
7.13 
log  3.57  =  0.5527 
log4.18  =  0.6212 
colog  7.13  =  9.1469  -  10 

0.3208 
=  log  2.093.   Ans. 


5.89  :  76.3  :  :  ? 
5.89  X  38.7 


38.7. 


=  ? 


76.3 
log  5.89  =  0.7701 
log38.7  =  1.5877 
colog  76.3  =  8.1175 


10 


0.4753 
=  log  2.987.    Ans. 

(3) 
7.37  :  ?  :  :  86.1  :  43.7. 
7.37  X  43.7  ^  ^ 
86.1 
log.7.37  =  0.8675 
log43.7  =  1.6405 
colog  86.1  =8.0650-10 

0.5730 
=  log  3.741.   Ans. 

(4) 
?  :  69.7  :  :  3.79  :  29.4. 
69.7x3.79^^ 
29.4 
log  69.7  =  1.8432 
log  3.79  =  0.5786 
colog  29.4  =  8.5317 -10 

0.9535 
=  log  8.984.    Ans. 


5.   Find,  by  logarithms,  the  values  of  0.08* 
^log   0.08  =  ^  of  (8.9031 -10)  =  9.6344 
J  log  2734  =  i  of  3.4368 
Jlog21.97  =  ^  of  1.3418 
3.6  log  7        =3.6x0.8451 


2734^;  21.97^;  7««. 

10  =  log  0.4309. 
1.1456  =  log  13.98. 
04478  =  log  2.801. 
3.0424  =  log  1103. 


446  ARITHMETIC. 


6.  Find,  by  logarithms,  the  values  of  9.71^ ;  7.935^ 

I  log  9.71    =  I  X  0.9872  =  2.3035  =  log  201.1.  (1)  Am. 
f  log  7.935  =  f  X  0.8996  =  0.6426  =  log  4.391.  (2)  Am. 

7.  What  is  the  horizontal  distance  between  two  points,  when  the 
air-line  distance  is  1534  ft.,  and  the  difference  of  level  34  ft.? 


V15342  -  342  =  V2352000  =  1533.623  ft.  Am. 

8.   Find  the  horizontal  distance  when  the  road  distance  is  1  mile, 
and  the  rise  347  ft. 


V(5280  +  347)(5280  -  347)  =  V27757991  =  5268.585  ft.  Am. 

9.  If  the  road  distance  is  half  a  mile,  and  the  horizontal  distance 
2513  ft.,  find  the  difference  of  level. 

V(2640  +  2513)(^40  -  2513)  =  V65443i  =  808.97  ft.  Am. 

10.  The  diagonal  of  a  rectangular  floor  is  34.6  ft.,  and  the  width 
is  17.8  ft.     Find  the  length  of  the  floor. 


V(34.6  +  17.8)(34.6  -  17.8)  =  V880.32  =  29.67  ft.  Am. 

11.  The  height  of  a  tower  on  a  river's  bank  is  55  ft.,  the  length 
of  a  line  from  the  top  to  the  opposite  bank  is  78  ft.  Find  the 
breadth  of  the  river. 


V(78  +  55)(78  -  55)  =  \/3059  =  55.31  ft.  Am. 

12.  The  number  of  seamen  at  Portsmouth  is  800,  at  Charlestown 
404,  and  at  Brooklyn  756.  A  ship  is  commissioned  whose  comple- 
ment is  490  seamen.  Determine  the  number  to  be  drafted  from  each 
place  in  order  to  obtain  a  proportionate  number  from  each. 

800  +  404  +  756  =  1960.  ^VW  X  ^p  =  101,  C. 

Wis  X  H^  =  200,  P.  ^^  X  ^  =  189,  B. 

13.  Show,  without  division,  that  36,432  contains  8,  9, 11  as  factors 

432  =  54  X  8. 
3  +  6  +  4  +  3  +  2=.  18. 

3  +  4  +  2  =  6  +  3.  (See  §  222.) 


teachers'  edition.  447 

14.  Find  the  smallest  multiplier  that  will  make  47,250  a  perfect 
cube. 

47,250  =  2  X  33  X  53  X  7. 
22  X  72  =  4  X  49  =  196.  Ans. 

15.  Find  the  proper  fraction  which,  when  reduced  to  a  continued 
fraction,  has  for  quotients,  1,  3,  5,  7,  2,  4. 

1  1«99     Ans. 


1  +  _L^      1443 
5  +  -     ' 


^-^-4 

^^i 


16.  If  the  meter  is  equal  to  1.09362  yds.,  find  a  series  of  four  frac- 
tions that  will  express  more  and  more  nearly  the  true  ratio  of  the 
meter  to  the  yard. 

1.09362  =  lTfM!Ty  =  1/oVoV 
4681)50000(10  ,.  i4e8.  =1+_1 


46810  ••  "5^0^^  10  1 


3190)4681(1  ^  ^         1 


3190 


1  + 


1491)3190(2  2  +  - 

2982  ' 

208)1491(7 
1456 
35 

1+1.11.  1+       1  =2^7. 

10     10  io  +  _JL_    235 

10  +  -       ^^  2  +  i 

1  7 

l  +  ^-r    =i-  ii.   i^.    ^,    ^-Ans. 

10  +  _I_     ^2  10'    11'    32    235 

^4 


448  ARITHMETIC. 


17.  Find  the  square  factors  contained  in  33,075. 

33075  =  33  X  5*  X  72.  32  X  5»  =  225, 

32  =  9,  32x7^  =  441, 

52  =  25,  52x72=1225, 

72  =  49.  32x52x72=11025. 

9,  25,  49,  225,  441,  1225,  11,025. 

18.  The  top  of  St.  Peter's,  Rome,  is  yf  ^  of  a  mile  above  the  ground, 
and  that  of  St.  Paul's,  London,  is  ^V?  of  a  mile.  By  how  many  feet 
does  the  height  of  St.  Peter's  exceed  that  of  St.  Paul's  ? 

20 

340  ft. 


48 
9    of  ^^^^-432  ft. 

;;p      1 

m^^ 

432  ft. -340  ft. 

=  92  ft.  Ans. 

19.  How  many  days  elapsed  between  the  annular  eclipse  of  May 
15,  1836,  and  that  of  March  15,  1858  ? 

1858  —  3  —  15  During  the  interval  there  were  five  leap  years, 

1836  —  5—15         and  in  the  ten  months  from  May  15  to  March  15 
01  _  iQ  there  are  304  days. 

365  X  21  =  7665.  ^665  +  304  +  5  =  7974  days.  Ans. 

20.  In  a  gale,  a  flag-staff  60  ft.  high  snaps  28.8  ft.  from  the  bot- 
tom ;  and,  not  being  wholly  broken  off,  the  top  touches  the  ground. 
If  the  ground  is  level,  how  far  is  the  top  from  the  bottom? 


The  distance  =  V(n72  +  28.8)(31.2  -  28.8)  =  V60  x  2.4 
=  \/l44"=  12  ft.  Ans. 

21.  Seventeen  trees  are  standing  in  a  line,  20  yds.  apart  from  each 
other ;  a  person  walks  from  the  first  to  the  second  and  back,  then  to 
the  third  and  back,  and  so  on  to  the  end.     How  far  does  he  walk  ? 

The  total  distance  is  the  sum  of  an  arithmetical  series  in  which 

n  =  16,   rf  =  40,   a  =  40. 
Z^  a  +  (n  -  l)(f  =  40  +  15  X  40  =  640. 
«  =  -  (a  +  0  =  ¥  X  (40  -h  640)  =  5440  yds.  =  3  mi.  160  yds.  Am. 


teachers'  edition.  449 


22.  A  level  reach  in  a  canal  is  14|  mi.  long  and  48  ft.  broad.  At 
one  end  is  a  lock  80  ft.  long,  12  ft.  broad,  and  with  a  fall  of  8  ft.  6  in. 
How  many  barges  can  pass  through  the  lock  before  the  water  in  the 
canal  is  lowered  1  in.  ? 

The  amount  of  water  that  can  be  drained  off  in  lowering  the 
level  1  in.  is  14|  x  5280  x  48  x  jV  =  311,520  cu.  ft.  The 
amount  of  water  wasted  each  time  a  barge  goes  through  the 
lock  is  80  X  12  X  8^  =  8160  cu.  ft.     Hence, 

311,520  ^  8160  =  38  barges.  Ans. 

23.  Find  the  capacity,  in  liters  and  in  bushels,  of  a  box  1.7™  long, 
g7cm  wide,  and  31««»  deep. 

1.7m_170cm_ 
V  =  0.908  qt. 
170  X  87  X  31  =  458,490«'™  =  458.491.  (1)  Am. 
458.49  X  0.908  -  416.309  qts. 
1  bu.  =  32  qts. 
416.309  ^  32  =  13  bu.  (2)  Ans. 

24.  Find  the  number  of  kilograms  of  olive  oil,  specific  gravity 
0.915,  to  fill  a  vessel  2.3""  long,  1.8""  wide,  and  74««»  deep. 

74cm  ^  0.74'". 
2.3  X  1.8  X  0.74  =  3.0636«i'n»  =  3063.6''k. 
3063.6''«  X  0.915  =  2803.194'^8.  Ans. 

25.  How  many  tons  in  a  block  of  marble  4  ft.  long,  34  in.  wide, 
17.3  in.  thick,  if  its  specific  gravity  is  2.73? 

4  ft.  =  48  in. 
48  X  34  X  17.3  =  28233.6  cu.  in. 
28233.6  ^  1728  =  16.34  cu.  ft. 
16.34  X  62i  lbs.  =  1021.25  lbs. 
1021.25  lbs.  X  2.73  =  2788.0125  lbs.  =  1.394  t.  Ans. 

26.  Find  the  surface  of  a  sphere  18.3  in.  in  diameter. 

The  radius  =  9.15  in. 
9.15  X  9.15  X  3.1416  =  263.0226  sq.  in. 

263.0226  X  4  =  1052.09  sq.  in.  Ans. 


450  ARITHMETIC. 

27.    Find  the  number  of  acres  in  a  circular  field  213  yds.  2  ft. 
across. 

213  yds.  2  ft.  =  04 1  ft.  in  diameter. 
The  radius  is  320.5  ft. 

1  A.  =  43560  sq.  ft. 

3.1416x320.52 


Area  = 


43560 


log  3.1416  =  0.4971 
log  320.5"^  =  5.01 16 
colog  43560  =  5.3609  -  10 

0.8696  =  log  7.407. 

7.407  A.  Am. 

28.  How  many  cubic  inches  in  a  10-inch  globe?  in  a  20-inch 
globe  ?     What  is  the  ratio  of  their  volumes  ? 

The  ratio  of  their  volumes  is  lO^ :  20^  -  l^ .  2^  =  1 :  8.  Am. 
10»  X  0.5236  =  523.6  cu  in.  (1).  Aiis 
523.6  X  8  =  4188.8  cu.  in.  (2).  Am. 

29.  How  many  balls  3  in.  in  diameter  can  be  ca«<t  from  a  pig  of 
iron  7  ft.  long,  6.7  in.  wide,  3.8  in.  thick,  if  the  waste  in  melting  and 
casting  is  reckoned  at  3^%? 

7  ft.  -  84  m. 

No.  of  balls  =  84  X  6.7  X  3.8x0.9675, 
3»  X  5236 

log        84=1.9243 

log       6.7  =  0.8261 

log       3.8  =  0.5798 

log  0.9675  =  9.9857 -10 
colog  3'  -  8.5686  -  10 
colog  0.5236  =  0.2810 

2.1655  =  log  146.4. 

Hence,  number  of  balls  is  146.  Aiis. 


teachers'  edition.  451 

30.  Find  the  diflference  in  length,  at  80°  F.,  of  a  glass  and  a  steel 
rod,  each  3  ft.  long  at  freezing  point,  if  the  expansion  at  100°  C.  is 
0.00085  for  glass  and  0.0012  for  steel. 

80°  F  =  f  (80°  -  32°)  C.  -  26 1°  C. 
Difference  in  length  at  100°  C.  is  0.0012  -  0.00085  =  0.00036. 
3  ft.  =  36  in. 
0.00035  X  36  X  0.26f  =  0.00366  in.  Ans. 

31.  A  grain  of  gold  is  beaten  out  in  leaf  to  cover  56  sq.  in.  "What 
weight  will  be  required  for  gilding  the  faces  of  a  cube  whose  edge 
is  3i  ft.  ? 

6x3^x31  =  731.  sq.ft. 
144  X  73^  sq.  ft.  ==  105.S4  sq.  in. 
10584  ^  56  =  189  grs.  =  7  dwt.  21  grs.  Ans. 

32.  What  premium  must  be  paid,  at  the  rate  of  1|%,  for  insuring 
a  vessel  worth  $117,750,  in  order  that  in  the  event  of  loss  the  owner 
may  receive  both  the  value  of  the  ship  and  the  premium  ? 

1-0.01875  =  0.98125. 
1117,750-0.98125  =  1120,000. 
$  120,000  -  117,750  =  $  2250.  Ans. 

33.  By  selling  goods  at  60  cts.  a  pound,  8%  on  the  cost  is  lost; 
what  advance  must  be  made  in  the  price  in  order  to  gain  15%  on 
the  cost? 

If  the  selling  price  is  92,  and  we  wish  it  to  be  115,  we  must  add 
23,  or  \  of  92.  Tliat  is,  wo  must  add  i  of  60  cts.  =  15  cts.  a 
pound.  Ans. 

34.  Divide  $27.12|^  among  three  persons,  giving  the  second  |5 
less  than  the  first,  and  twice  as  much  as  the  third. 

Let  a  represent  the  ainotmt  given  to  the  first. 

a  —  5  will  represent  the  amount  given  to  the  second. 

a     5 

will  represent  the  amount  given  to  the  third. 

Then,  2\a  —  *l\  will  represent  the  amount  given  to  all  three. 


452  ARITHMETIC. 


But  $27.12^  is  the  amount  given  to  all  three. 
Hence,  2\a -1\  =  21.\2\. 

Add  to  each  side  7i  2^a  =  34.625 

Divide  each  side  by  2^,  a  =  13.85 

a  -  5  =  8.85 
Ka- 5)  =  4.421 
Hence,  1st  gets  1 13.85;  2d,  $8.85;  3d,  $4.42^.  Am. 

35.  The  population  of  a  city  in  1880  was  12,298,  showing  a 
decrease  of  8|%  on  its  population  in  1870;  in  1870  there  was  an 
increase  of  7|%  on  the  census  of  1860.  What  was  its  population 
in  1860? 

12,298  H-  0.91f  =  13,416,  population  in  1870. 
13,416  H-  1.075  =  12,480,  population  in  1860.  Am. 

36.  Find  the  increase  of  income  obtained  by  transferring  |2500 
from  3%  stocks  at  94^  to  4%  stocks  at  105. 

0.03  of  $2500  =  $75,  income  from  the  3%  stock. 

0.94^  of  $2500  .-=  $2362.50,  amount  from  the  3%  stock. 

$  1.05  is  paid  for  $  1  worth  of  4%  stock. 

Hence,  $  2362.50  is  paid  for  2362.50  -f- 1.05  =  $  2250  stock. 

0.04  of  $2250  =  $90,  income  from  4%  stock. 

$90  —  $  75  =  $  15,  increase  of  income.  Am. 

37.  Each  person  breathing  in  a  closed  room  spoils  the  air  at  the 
rate  of  about  8  cu.  ft.  a  minute.  A  congregation  of  400  persons 
enter  a  closed  room  70  ft.  by  40  ft.  and  20  ft.  high.  How  long  will 
it  take  them  to  spoil  the  air  ? 

70  X  40  X  20 

The  air  in  the  room  will  last  one  person  - — — —  min.,  or 

400  persons, 

M|1.^.17i„.in.^n.. 

^xm      2      ^ 

2      Xii 

38.  How  long  can  the  windows  and  doors  of  a  school-room  be 
safely  kept  closed  when  occupied  by  50  children,  if  the  room  is 
25  ft.  by  20  ft.  and  10  ft.  high  ? 


teachers'  edition.  453 


25  X  20  X  10 
The  air  in  the  room  will  last  one  person min.,  or 

50  persons, 

^X^0  2  ' 

2      ^ 

39.  Find  the  square  root  to  four  decimal  places,  of  the  reciprocal 
of  0.0043. 

^  colog  0.0043  =  1.1833  =  log  15.25.  Ans. 

40.  A  pays  B  $230  as  the  present  value  of  |300  due  in  5  yrs. 
Which  gains  by  the  payment,  and  how  much,  if  interest  is  reckoned 
at  50/0? 

The  present  worth  of  |300  due  in   5   yrs.   is      }^^^     of  $300 

=  $235.05.     Hence,  A  gains  $5.05  by  the  transaction. 

41.  Find  the  quantity  of  coal  required  by  a  steamer  for  a  voyage 
of  4043  mi.,  if  her  rate  per  hour  is  14.04  knots,  and  her  consumption 
of  coal  87  t.  per  day. 

14.04  X  6086  X  24  =  2,050,738.56  ft.  per  day. 
4043  X  5280  =  21,347,040  ft. 
21,347,040  -i-  2,050,738.56  =  10.40944  days. 

10.40944  X  87  t.  =  905  t.  1392  lbs.  Ans. 

42.  Find  the  area  of  a  circular  ring  of  which  the  inner  and  outer 
diameters  are  7.36  and  10.64  in. 

Area  =  0.7854(10.642  -  7.362) 

=  0.7854  (113.2096  -  54.1696) 
=  0.7854  X  59.04 
=  46.37  sq.  in.  Ans. 

43.  A  and  B  can  do  a  piece  of  work  in  13^  dys.,  A  and  C  in  10| 
dys.,  A,  B,  and  C  in  7^  dys.     In  how  many  days  can  A  do  it  alone  1 

In  1  day  A  and  B  can  do  ^^  of  the  work. 
In  1  day  A  and  C  can  do  /^  of  the  work. 


454 


ARITHMETIC. 


In  1  day  A,  B,  and  C  can  do  ^y  of  the  work. 
.".  in  1  day  B  can  do  ^^  —  /^  of  the  work. 
And  in  1  day  C  can  do  ^  —  :^^  of  the  work. 
And  in  1  day  A  can  do  ^^  —  (^  —  3^)  of  the  work. 

3      /  2       3  \  _  3       2       3  _  36  -  64  +  45      17 
40     Vl5     32/     40     15     32  480  480* 

.-.  it  will  take  A  -Vr  =  28^^  dys.  Ans. 


44.  If  3  men  working  11  hrs.  a  day  can  reap  20  A.  in  11  dys,, 
how  many  men  working  12  hrs.  a  day  can  reap  a  field  360  yds.  long 
and  320  yds.  broad  in  4  dys.  ? 

360  yds.  X  320  yds.  =  115,200  sq.  yds. 
115,200  H-  30i  -5-  160  =  ^^/Ji  A. 

3 

n 

m 
3 : what ^       n^n^%m^?> ^ ^ ,,„.  ^,, 

n 


12 

11 

4 

11 

20 

¥^¥ 

45.   Find  the  area  of  a  triangle  whose  sides  are  12,  5,  and  13  in. 

Observe  that  13^  =  12-'  +  5".     Hence  the  triangle  is  a  right  tri- 
angle.       Area  =  J^  of  12  X  5  =  30  sq.  in.  An&, 


46.    Find  the  area  of  a  triangle  whose  sides  are  73,  57,  and  48  ft. 
53  +  57  +  48  ^  gc^ 


Area  =  V80  x  16  x  32  x  41. 

log  89  =1.9494 
log  16 -1.2041 
log  32 -1.5051 
log41-.  1.6128 

2)6.2714 

3.1357  =  log  1367. 
1367  sq.  ft.  An%. 


teachers'  edition.  455 


47.    Find  the  number  of  hektars  in  a  triangular  field  whose  sides 
are  37.5°S  91.7°',  and  78.9'". 

3r^+^L7  +  7M^  104.05. 


Area  =  V104.05  x  66.55  x  12.35  X  25.15. 

log  104.05  =  2.0172 
log  66.55  =  1.8232 
log  12.35  =  1.0917 
log   25.15  =  1.4006 


2)6.3327 
3.1663  =  log  1467. 
1467i"'  =  0.1467h^  Ans. 


48.   Find  the  number  of  hektars  in  a  triangular  field  whose  sides 
are  67.5°^,  81.2'",  and  102.7'". 

67.5  +  81.2  +  102.7  _  -j^g  7 


Area  =  V125.7  X  58.2  x  44.5  x  23 

log  125.7  =  2.0994 
log  58.2=1.7649 
log  44.5  =  1.6484 
log   23    =1.3617 

2)6.8744 
3.4372  =  log  2736. 
27361'°  =  0.2736ha.  Ans. 


49.   Find  the  number  of  acres  in  a  triangular  field  whose  sides  are 
227,  342,  and  416  ft. 

1  A.  =  43,560  sq.  ft. 

227jl342Jl416^492  5. 

2 

Area  =  ^^^2.5  X  265.5  x  150.5  x  76~5  ^ 
43560 


456  ARITHMETIC. 


log  492.5  =-  2.6924 
log  265.5  =  2.4241 
log  150.5  =  2.1776 
log   76.5  =  1.8837 

2)9.1778 
4.5889 
colog  43,560  =  5.3609 -10 


9.9498 -10  =  log  0.8908. 
0.8908  A.  Ans. 


50.   Find  the  number  of  acres  in  a  triangular  field  whose  sides  are 
79  chains  8  links,  57  chains  3  links,  and  102  chains  19  links. 

79.08  +  57.03  +  102.19  _  ^^g  ^^ 


Area 


2 
Vl  19.15  X  40.07  X  62.12  X  16.96 


10 

log  119  15.=  2.0761 
log  40.07  =  1.6028 
log  62.12=1.7932 
log   16.96  =  1.2294 

2)6.7015 
3.3508 
colog  10  =  9.0000  -  10 

2.3508  =  log  224.3.        224.3  A.  Am. 

51.    Find  the  number  of  square  rods  in  a  triangle  whose  sides  are 
7  rds.  2  yds.,  6  rds.  5  yds.,  and  9  rds.  4^  ft. 

7  rds.  2  yds.  =  121.5  ft. 
6  rds.  5  yds.  =  114  ft. 
9  rds.  4i  ft.  =  153  ft. 

1  rd.  =  272.25  sq.  ft. 
121.5  +  114  +  153     ^g^og 
2 

Area  =  V194.25  x  72.'r5  X  ft0.2f^  Y IT^ 
272.25  ' 


teachers'  edition.  457 


log  194.25  =  2.2884 

log    72.75  =  1.8618 

log   80.25=1.9043 

log  41.25  =  1.6155 

2)  7.6700 

*   3.8350 

colog  272.25  =  7.5650  -  10 

1.4000  =  log  25.12.        25.12  sq.  rds.  Am. 

52.  Find  the  number  of  acres  in  a  four-sided  field,  the  sides  of 
which  are  in  order  361,  561,  443,  and  357  ft. ;  and  the  distance  from 
the  beginning  of  the  first  side  to  the  end  of  the  second  side  is  682  ft. 

361  +  561  +  682 


2 


802. 


Area  =  ^§0^  X  441  x  241  x  120  ^ 
43560 
log  802=  2.9042 
log  441=  2.6444 
log  241=  2.3820 
log  120=  2.0792 
2)10.0098 
5.0049 
colog  43560=  5.3609  -  10 

0.3658  =  log  2.322. 


357  +  443  +  682  _  ^^^ 

2 


Area  =  V741  x  384  X  298  X  59 
43560 
log  741  =  2.8698 
log  384  =  2.5843 
log  298  =  2.4742 
log  59  =  1.7709 
2)9.6992 
4.8496 
colog  43560  =  5.3609  -  10 

0.2105  =  1.624 

1.624  +  2.322  =  3.946  A.  Ans. 


458  ARITHMETIC. 


53.  Find  the  number  of  hektars  in  a  field  of  three  sides,  one  of 
wliich  is  82.1™,  and  the  distance  from  this  side  to  the  opposite  corner 
is  47.3'». 

^  (82.1  X  47.3)  =  1941.671°'  =  0.194167^'^  Aiis. 


54.  Find  the  number  of  acres  in  a  triangular  lot,  one  side  of  which 
is  313.6  ft.,  and  the  distance  from  this  side  to  the  opposite  corner  is 
163.2  ft. 

J  (343.6  X  163.2)  =  28037.76  sq.  ft. 
28037.76  -5-  43560  =  0.6436  A.  Am. 


55.    Find  the  altitude  of  a  triangle,  if  each  side  is  1000  ft 
1000  +  1000  +  1000  _  ^5QQ 


Area  =  V1500  x  500  x  500  x  500 
=  250000  \/3  =  43301 2.5  sq.  ft. 
433012.5  ^  500  =  866.025  ft.  Ans. 

56.    Find  the  distances  of  the  vertices  from  the  opposite  sides  of  a 
triangle,  when  these  sides  are  17.8"^'",  23.6"'",  and  31.5°>"». 

17.8  +  23.6  +  31.5  _  ^^^^ 


Area  =  V36.45  x  18.65  x  12.85  x  4.95. 

log  36.45  =  1.5617  log  area=  2.3179 

log  18.65  =  1.2707  colog  11.8  =  8.9281  -  10 

log  12.85  =1.1089  JTTTT 

log   4.95  =  a694_6  =  log  17.62. 

2)4.6359  17.62»».  (2)  Ans. 

log  area  =  2.3179  log  ^rea  =  2.3179 

colog  8.9  -  9.0506  -  10  colog  15.75  =  8.8027  -  10 

1.3685  1.1206 

-=  log  23.36.  =  log  13.2. 

23.36""".  (1)  Ans.  13.2"'™.  (3)  Ans. 


teachers'  edition.  459 

57.  If  the  four  sides  of  a  field  measured  in  succession  are  237,  253, 
244,  and  261  ft.,  and  the  diagonal  measured  from  the  end  of  the  first 
side  to  the  end  of  the  third  side  is  351  ft. ;  find  its  area. 

237  +  261  +  351  _  ^^^  ^ 
2  ■  ' 

Area  of  triangle  =  V424.5  x  187.5  x  163.5  x  73.5. 

log  424.5  =  2.6279  log  424  =  2.6274 

log  187.5  -  2.2730  log  171  =  2.2330 

log  163.5  =  2.2135  log  180  =  2.2553 

log    73.5  =  1.8663  log   73  =  1.8633 

2)8.9807  2)8.9790 

4.4904  4.4895 

=  log  30,925.  _  log  30,860. 
253  +  244  +  351 


424.  30,860  +  30,925  =  61,785  sq.  ft 

2  ,  .  .         H 


Area  of  triangle 


ns. 


\/424x  171x180x73. 


58.  If  the  four  sides  of  a  field  are  237,  253,  244,  and  261  ft.,  taken 
in  order,  and  if  the  corner  formed  by  the  second  and  third  sides  is  a 
square  corner  ;  find  the  diagonal  from  the  beginning  of  the  second 
side  to  the  end  of  the  third  side,  and  also  find  the  area  of  the  field. 


Diagonal  =  \/253'^  +  244^  =  VT23545  =  351.489. 
The  area  is  the  same  as  in  the  last  problem. 

59.  Find  the  area  of  a  circle  that  has  a  radius  of  10  in.  ;  of  a  cir- 
cle that  has  a  diameter  of  10  ft.  ;  of  a  circle  that  has  a  circumference 
of  30  in. 

Area  =  irR'  =  3.1416  x  100  =  314.16  sq.  in.  (1)  Ans. 
Area  =  rri^^  =  3.1416  x  25  =  78.54  sq.  ft.  (2)  Ans. 
Circumference  =  2  irE. 
Z0  =  2irR. 

R  =  ^± 

IT 

Area  =  -jriS^  =  i:  x  —  =  -^^  =  71.62  sq.  in.  (3)  Ans. 
1       TT^       3.1416  ^ 


460 


ARITHMETIC. 


60.  A  horse  is  tied  by  a  rope  28.8™  long ;  what  part  of  a  hektar 
can  he  graze? 

Area  =  ttE"  =  3.1416  X  27.82  =  3.1416  x  772.84 
=  2427.951'"  =  0.2428'>».  Ans. 

61.  How  many  square  feet  in  a  circle  that  has  a  diameter  of  17f 
yds.? 

17f  yds.  =  53  ft. 

Area  =  itB:^  =  3.1416  x  26.5'. 
log    26.52  =.  3.8464 
log  3.1416  =  0.4971 

3.3435  =  log  2205.5 
2205.5  sq.  ft.  Am. 

62.  How  many  square  feet  in  a  circle  that  has  a  circumference  of 
117  yds.? 

log  3512  =  5.0906 
colog  3.1416  =  9.5029 -10 
colog4  =9.3979-10 

3.9914 
=  log  9804. 
9804  sq.  ft.  Am. 


117 

yds. 

=  351  ft. 

Circumfei 

ence 

==2-nR. 

351 

=  2nB. 

R 

_351 
27r" 

»i22 

35P 

4  X  3.1416 

63.  How  many  square  inches 
in  the  surface  of  a  globe  that  has 
a  radius  of  12.37  in.? 

Area  =  4  tt/?* 

=  47rx  12.37*. 
log  4  =  0.6021 

log  3.1416  =  0.4971 
log  12.37^=2.1848 

3.2840  -  log  1923. 
1923  sq.  in.  Am. 

64.  Find  the  area  of  the  sur- 
face of  the  largest  globe  that  can 


be  turned  out  (Voni  a  joist  4  in. 
by  6  in. 

Area  =  4  7r22-16T 

=  1<;  •  :'..l  \\r> 

=  50.Jtif>  S^J.    111.      .I/(N. 

65.  How  many  cubu-  liuin-,-^ 
in  a  globe  that  has  a  diameter  of 
10  in.? 

Volume  =  0.5236  x  diam.» 
=  0.5236x1000 
—  523.6  sq.  in.  Am. 


TEACHERS     EDITION. 


461 


66.  If  a  tree  be  round,  and 
the  girt  is  17  ft.  6  in.,  find  its 
diameter.  Find  the  area  of  a 
cross-section,  and  find  the  num- 
ber of  cubic  feet  in  the  largest 
sphere  that  can  be  cut  from  it. 

Diameter  =  - — ~ 
3.1416 
=  5.57  ft.  (1)  Ans. 
Area  =  Tri?^. 
log  3.1416  =  0.4971 
log  5.572    =1.4918 
colog  22  =  9.3979  -  10 
log  Area  =  1.3868  =  log  24.37. 

24.37  sq.  ft.  (2)  Ans. 
Volume  =  0.5236  x  diam.^ 
log  0.5236  =  9.7190 -10 
log  5.573    =  2.2377 

1.9567  =  log  90.52. 
90.52  cu.  ft.  (3)  Ans. 

67.  Find  the  weight  in  kilo- 
grams and  in  pounds  of  an  iron 
ball  21.5''™  in  diameter,  specific 
gravity  7.47 ;  of  a  tin  ball  13<=°» 
in  diameter,  specific  gravity  7.29  ; 
of  a  lead  ball  17.3'='»  in  diameter, 
specific  gravity  11.35  ;  of  a  silver 
ball  1.31°™  in  diameter,  specific 
gravity  10.47. 

Iron. 
V=  0.5236  X  21.53 

log  0.5236  =  9.7190 -10 
log  21.53     =3.9972 
log  747     =  0.8733 
colog  1000  =  7.0000-10 

1.5895  =  log  38.86. 
38.86''8.  (1)  Ans. 
log  2.205  .  =  0.3434 
log  38.86    =  1.5895 

1.9329  =  log  85.68. 
85.68  lbs.  (2)  Ans. 


Lead. 

log     17.33  =  3.7140 
log  0.5236  =  9.7190 -10 
log    11.35  =  1.0550 
colog  1000  =  7.0000  -  10 

1.4880  =  log  30.76. 
30.76''st.-(l)  ^ns. 

log   2.205  =  0.3434 
log   30.76  =  1.4880 

1.8314  =  leg  67.83. 
67.83  lbs.  (2)  Ans. 


Tin. 

F=  0.5236  X  133. 
log  0.5236  =  9.7190 -10 
log       133  =  3.3417 
log     7.29  =  0.8627 
colog  1000  =  7.0000-10 

0.9234  =  log  8.383. 
8.383'^g.  (1)  Ans. 

log  2.205  =  0.3434 
log  8.383  =  0.9234 

1.2668  =  log  18.48. 
18.48  lbs.  (2)  Ans. 

Silver. 

log    1.313  =  0.3519 
log0.5236  =  9.7190 -10 
log    10.47  =  1.0199 
colog  1000  =  7.0000  -  10 
8.0908  -  10 
=  log  0.01233. 


0.01233'^?.  (1)  Ans. 


log      2.205 
log  0.01233 


0.3434 
8.0908  -  10 
10 


8.4342 
=  log  0.0271 7. 
0.02717  lbs.  (2)  A71S. 


462  ARITHMETIC. 


68.  A  slab  of  cast-iron  4  ft.  2^  in.  long,  17  in.  wide,  and  8^  in. 
thick,  specific  gravity  7.31,  is  cast  into  2-lb.  balls.  If  there  is  a  loss 
of  5%  in  melting,  how  many  balls  are  obtained,  and  what  is  the 
diameter  of  each  ? 

The  Blab  will  make  50.5x17x25x0.95x62.5x7.31  ^^^^ 

2  X  3  X  1728 
log  50.5  =  1.7033  The  diameter  will  be 

^      oI^J?o^o  i/50.5x  17X25X0.95 

log     25=1.3979  ^     0.5236X3X698 

log  0.95  =  9.9777 -10 

log  62.5  =  1.7959  ^^g     50.5  =  1.7033 

log  7.31  =  0.8639  ^  ^^g        17=1.2:m 

colog       2  =  9.6990-10  ^^8        25=1.3979 

colog       3  =  9.5229-10  ^^g     0.95  =  9.9777-10 

colog  1728  =  6.7625  -  10  ««^«g  ^-^^236  =  0.2810 

^  colog  3  =  9.5229-10 

2.9535  =  log  898.  ^olog      898  =  7.0467  -  10 

898  balls.  (1)  Ans.  3)1.1 5v>9 

0.3866 
=  log  2.436. 
2.436  inches.  (2)  Arts, 

69.  How  many  pounds  avoirdupois  would  a  ball  of  such  iron 
30  in.  in  diameter  weigh  ? 

log       3a''  =  4.4313 
log0.5236  =  9.7190 -10 
log     62.5  =  1.7959 
log     7.31  =  0.8639 
colog     1728  =  6.7625-10 

3.5726  =  log  3732.5. 
3732.5  lbs.  Am. 


70.  If  the  specific  gravity  of  ice  is  0.921,  find  the  weight  and  the 
surface  of  each  of  three  spheres  of  ice  whose  diameters  are  l***,  10"™, 
and  l"*.  Which  of  these  spheres  would  roll  first  on  a  plain,  in  a 
gradually-increasing  wind? 


teachers'  edition.  4G3 

F=  P  X  0.5236  -  0.5236««'«. 

0.5236«<='»  =  523. e^K. 
0.921  X  523.6™«  =  482.24'ng   (1)  Ans. 
V=  103  X  0.5236  =  523.6««'». 
523.6««°»  =  523.68. 
0.921  X  523.6s  =  482.248.  (2)  Am. 
V=  13  X  0.5236  =  0.5236«*>m^ 

0.5236«bm  =  523.6»'8. 
0.921  X  523.6''8  =  482.24'^8.  (3)  Ans. 
Area  =  AirE"  =  irD'' =  3.1416  x  P     =  3.14161'=™    (1)  Ans. 
Area  =  4  7r/?'''  =  irD'  =  3.1416  x  10^    =  314.16<i«'».  (2)  Ans. 
Area  =  AtR'  =  ttW  =  3.1416  x  100^  =  31,416q«n».  (3)  Ans. 
The  first,  if  we  take  no  account  of  friction.  (4)  Ans, 

11.  Given  a  cylinder  10  in.  in  diameter  and  12  in.  long;  required 
the  area  of  each  end,  the  convex  area,  the  total  area,  and  the  contents 
in  gallons. 

Area  of  end  =  3.1416  x  5^  =  3.1416  X  25  =  78.54  sq.  in.  (1)  Ans. 
Convex  area  =  tt  X  10  x  12  =  3.1416  x  120  =  376.99  sq.  in.  (2)  Ans. 
Total  surface  =  376.99  +  2  x  78.54  =  534.07  sq.  in.  (3)  Ans. 
1.02       4 
F=  ^^-^^  ^  ^^  gals.  =  4.08  gals.  (4)  Ans. 

n 

72.  Find  the  capacity  in  gallons  of  a  round  cistern  13  ft.  in 
diameter  and  9  ft.  deep. 

Tr     3.1416  X  6.52  X  9  X  1728 


231 

log  3.416  =  0.4971 
log  6.52  =1.6258 
log  9  =  0.9542 
log  1728  =3.2375 
colog  231  =  7.6364 -10 

3.9510  =  log  8935. 
8935  gals.  Ans. 


464  ARITHMETIC. 


73.  What  must  be  the  diameter  of  a  cylinder  10  in.  deep,  in  order 
that  it  may  hold  1  gallon  ? 

V=^irR'xh.         231  =  Ti?»xlO.         i2==J^. 

AflOr 
log  231       =2.3636 
colog  10        =  9.0000  -  10 
colog3.1416  =  9.5029 -10 

2)0.8665 
0.4333  =  log  2.712. 
2.712  in.  X  2  =  5.424  in.  Ans. 

74.  Find  the  volume  of  a  cylinder  8  in.  in  diameter  and  11  in. 
high. 

V=itx  4'^  X  11  =  3.1416  X  16  X  11  =  552.92  cu.  in.  Am. 

I 

75.  Find  the  dimensions  of  three  cylinders  that  have  the  diam- 
eters equal  to  the  heights,  and  hold  1  gal.,  1  qt.,  and  1^  respectively. 

log  231  =2.3636 
colog  3.1416  =  9.5029  -  10 

3)1.8665 

0.6222  =  log  4.19. 
4.19  in.  (2)  Am. 


F=^. 
4 

11  =.  lOOOoon 
1000  =  1:^. 

4 


F=  irR'h 

='(fy-f- 

231 

4 

D 

' /4  X  231 
>»  3.1416* 

log     4 

=  0.6021 

log  231 

=  2.3636 

colog  3. 14 16 

=  9.5029  - 10 
3)2.4686 

0.8229=  log 

6.651 

6.65 

in.  (1)  An&. 

V 

4 

Iqt. 

=  -2|icu.  in. 

Z^X 

=  irZ)». 

D 

=;'23i. 

>'3.1416 

n^'i 


4000 


3.1416 
log  4000  =  3.6021 
colog  3.1416  =  9.5029 

3)3.1050 
1.0350  =  log  10.84. 
10.84«».  (3)  Am. 


teachers'  edition.  465 


76.  Find  the  volume  of  a  triangular  priam  11  in.  long,  the  sides 
of  the  ends  being  2,  3,  and  4  in.  long. 

Area  of  end  =  V4.5  x  2.5  x  1.5  x  0.5  =  2.9047. 
F=  11  X  2.9047  =  31.952  cu.  in.  Ans. 

77.  Find  the  capacity  in  bushels  of  a  bin  6  ft.  long,  the  end  of 
which  is  a  square  measuring  3  ft.  3  in.  on  a  side. 

F=  6  X  3^  X  31  =  6  X  \'  X  -\«  =  -5-F  =  63f  cu.  ft. 
0.80356  bu.  =  1  cu.  ft. 
0.80356  X  63^-  =  50.93  bu.  Ans. 

78.  Find  the  number  of  cubic  yards  in  a  square  prism  200  ft.  on  a 
side,  and  40  ft.  long. 

J.     2002  y^  40 

V= —  =  592o9^'y  cu.  yds,  Ans. 

79.  How  many  cubic  yards  in  a  square  pyramid  210  ft.  on  a  side, 

and  123  ft.  high  ? 

210  ft.  =  70  yds. 
123  ft.  =  41  yds. 
V=  i(702  X  41)  =  669661  cu.  yds.  Ans. 

80.  Find  the  capacity  of  a  cup,  the  mouth  of  which  is  a  square 
4  in.  on  a  side,  and  the  sides  of  which  are  four  equilateral  triangles. 


^  the  diagonal  of  base  =  |\/l6  +  16  =  |  V32. 

Altitude  =  V42-(i>/32)2  =  Vl6^^  =  VS  =  2.8285. 

V=  i  of  16  X  2.8285  =  15.085  cu.  in.  Ans. 

81.  The  largest  of  the  Egyptian  pyramids  is  14 T*"  high,  with  a 
base  231™  square.     Find  its  volume  in  cubic  meters. 

^  of  23P  X  147  =  2,614,689°'"".  Ans. 

82.  The  slant  depth  of  a  conically-shaped  drinking-cup  is  93°»"*, 
and  the  diameter  at  the  top  8«°».     What  is  its  capacity  ? 


466  ARITHMETIC. 


93min  ^  9_3cm_ 


Height  =  V9.32  -  4'  -  S.-SeSS^"*. 
F=|tX42x8.3958°"» 
=  ^  of  3.1416  X  16  X  8.3958  =  140.67«'°'  =  0.1406?.  Ans. 

83.   Tlie  volume  of  a  cone  is  1"**™  ;  its  height  is  equal  to  the  radiiis 
of  its  base.     Find  the  dimensions  of  the  cone. 


y  _7r  B^h  v=—  E^  =  ^  ^_     /3_x_1000000 

log3,000,000  =  6.4771 
colog  3.1416  =  9.5029 -10 

3)5.9800 

1.9933  =  log  98.47. 
98.47«".  Ans. 

84.  Find  the  capacity  of  a  wash-bowl  30*^  in  diameter  and  5*"' 
deep. 

i  of  30»  =  J  of  900  =  225. 
J  of   52  =  J  of    25  =  8.33. 

225  +  8.33  =  233.33. 
-»/  of  5  X  233.33  =  1833.31<'«»'  =  1.833'.  Am. 

85.  Find  the  capacity  in  liters  of  a  boiler  89«™  in  diameter  and 
31''"»  deep. 

i  of  892  =  J  of  7921  =  1980.25. 
iof3P  =  iof   961  =  320.33. 
1980.25  +  320.33  =  2300.58. 
V  of  31  X  2300.58  =  112,071. ll^""  -  112.071.  Ans. 

86.  Find  the  capacity  in  quarts  of  a  bowl  10  in.  in  diameter  and 
4  in.  deep. 

}  of  10»  =  i  of  100-25. 
J  of   4'»  =  iof    16  =  5.333. 
25  +  5.333  =  30.333. 
V  of  4  X  30.333  =  190.66  cu.  in. 
1  qt.  —  57.75  cu.  in. 
190.66  +  57.75  =-  3.3  qt«.  Ans. 


teachers'  edition.  467 


87. 

deep;  of  a  bowl  7  in.  across  and  3  in.  deep;  of  a  bowl  8  in.  across 
and  3j  in.  deep. 

1  pt.  =  I  of  231  cu.  in.  =  28.875  cu.  in. 
Jof  62  =  1  of  36  =  9. 
i  of  (11)2--=^  of  1  =  0.75. 
9  +  0.75  =  9.75. 
-If  of  1.5  X  9.75  =  22.982  cu.  in. 
1  pt.  =  28.875  cu.  in. 
22.982  -^  28.875  =  0.8  pt.  (1)  Ans. 

J  of  72  =  i  of  49  =  12.25. 
i  of  32  =  ^  of   9  =  3. 

12.25  +  3  =  15.25. 
-V-  of  3  X  15.25  =  71.893  cu.  in. 
71.893  -  28.875  =  2.5  pts.  (2)  Ans. 


J  of  82  =  J  of  64  = 

=  16. 

Jofa)2  =  iof-V-  = 

=  4tV 

16  +  4tV  = 

=  20tV 

i 

n 

x?x24lx  ^  -241  = 

=  3.8  pts.  (3) 

Ans. 

7 

^    n    m    63 
3     21 

X^            V     / 

• 

88. 

How 

many  gallons  will  a  boiler  5  ft.  in 

diameter  and 

deep 

hold? 

^  of  52  =  ^  of  25  = 

iof22  =  iof   4  = 

6.25  +  1.333  = 

=  6.25. 
=  1.333. 
-  7.583. 

V  of  2  X  7.583  = 

=  23.832  cu.  ft. 

23.832  X  1728  -^  231  = 

=  178.3  gals. 

Ans. 

2  ft. 


89.    How  many  gallons  will  a  boiler  30  in.  in  diameter  and  1  ft. 
deep  hold  ? 

\  of  302  _  ^  of  900  =  225. 
i  of  122  =  ^  of  144=    48. 
225  +  48  =  273. 
-Vof  12x273  =  5148  cu.  in. 
5148  ^  231  =  22.3  gals.  Ans. 


468  ARITHMETIC. 


90.  Find  the  capacity  in  pints  of  a  cylinder  1.9375  in.  in  diameter, 
2.4375  in.  high ;  of  a  cylinder  3^  in.  in  diameter,  3|  in.  high;  of  a 
cylinder  3||  in.  in  diameter,  5^^  in.  high. 

1  pt.  =  28.875  cu.  in. 

V=  ^ilil^  X  0-9fi875g  X  2.4375 
"28.875 

log    3.1416  =  0.4971 
log  0.968752  =  9.9724 -10 
log    2.4375  =  0.3870 
colog     28.875  =  8.5394 

9.3959 -10  =  log  0.2488. 
0.249  pt.  (1)  Ans. 

y^  3.1416  X  1.5625'  x  3.625 


log  3.1416  =  0.4971 
log  1.56252  =  0.3876 
log    3.625  =  0.5593 
colog  28.875  =  8.5394  -  10 


9.9834  = 

log  0.9625. 

0.963  pt. 

(2)  Alls. 

3.1416  X 

:  1.90625' 

X5.06 

28.875 

3.1416  = 

=  0.4971 

F= 

log 

log  1.906252  =  0.5605 
log    5.0625  =  0.7044 
colog     28.875  =  8.5394  - 10 

0.3014  =  log  2.002. 
2.002  pt.  (3)  Am. 

91.  Find  the  capacity  in  pecks  of  a  cylinder  15.865  in.  in  diameter, 
12.5  in.  high  ;  of  a  cylinder  9.25  in.  in  diameter,  4.25  in.  deep ;  of  a 
cylinder  18.5  in.  in  diamet<3r,  8  in.  deep. 

y^  0.7854  X  15.865'  x  12.5  x  4 
2150.42 


teachers'  edition.  469 

log   0.7854  =  9.8950-10 
log  15.8652  =  2.4009 
log       12.5  =  1.0969 
log  4  =  0.6021 

colog  2150.42  =  6.6675  -  10 

0.6624  =  log  4.596. 

4.596  pks.  (1)  Ans. 

y^  0.7854  X  9.25^  x  4.25  X  4 
2150.42 

log   0.7854  =  9.8950-10 
log      9.252  =  1.9322 
log       4.25  =  0.6284 
log  4  =  0.6021 

colog  2150.42  =  6.6675 -10 

9.7252 -10  =  log  0.531. 

0.531  pks.  (2)  A71S. 

y^^0.7854x  18.52x8x4 
2150.42 

log   0.7854  =  9.8950-10 
log      18.52  _  2.5344 
log  8  =  0.9031 

log  4  =  0.6021 

colog  2150.42  =  6.6675  -  10 

0.6021=  log  4. 

4  pks.  (3)  Ans. 

92.   What  must  be  the  diameter  of  a  circle,  in  order  that  it  may- 
contain  78.54  sq.  ft.  ?  to  contain  314.16  sq.  ft.  ? 

Area  =  0. 7854  x  D''.  Area  =  0. 7854  x  D^. 

78.54  =  0.7854  x  D\  314.16  =  0.7854  x  D\ 

7)2=100.  i)2_4oo. 

i>  =  10  ft.  (1)  Ans.  i)  =  20  ft.  (2)  Ans. 


470  ARITHMETIC. 


93.   What  must  be  the  diameter  of  a  circle  to  contain  1  A.  ?  to 
contain  9  A.  ? 


1  A.  =  43,560  sq.  ft.  jy  _     /43560 

Area  =  0.7854  X  D^.  \0.7854 

log  43,560  =  4.6.391 
colog  0.7854  =  0.1049 

2)4.7440 

2.3720  =  log  235.5. 

235.5  ft.  (1)  Am. 
V9  =  3.  3  X  235.5  =  706.5  ft.  (2)  Am. 

94.    What  must  be  the  diameter  of  a  circle  to  contain  l***?  to  con- 
tain 25h«  •' 


l**"^  =  10,0001™.  r^     .  <  10000 


.7854 
log  10.000  =  4.0000 
colog  0.7854  =  0.1049 

2)4.1049 

2.0525  =  log  112.8. 

112.8"'.  (1)  Am. 
\/25  =  5.  5  X  112.8"'  =  564"'.  (2)  Am. 

95.  Find  the  number  that  exceeds  its  square  root  by  20. 

On  testing  the  square  numbers  exceeding  20,  namely,  25,  36, 
etc.,  we  see  that  25  is  the  number  ;  and  no  process  of  approxi- 
mation is  needed.  25,  Am. 

96.  How  much  water  will  a  hemispherical  bowl  hold  that  is  10 
in.  in  diameter  ? 

V=  \  of  0.5236  X  2>»  =  1  of  0.5236  x  1000  =  261.8  cu.  in.  Am. 

97.  What  will  it  cost  to  gild  a  hemispherical  dome  10  ft.  in  diam- 
eter, at  50  cents  a  square  foot  ? 

8=-2x  0.7854  x  10»  =  157.08  sq.  ft. 
157.08  X  $0.50  =  $78.54.  Am. 


teachers'  edition.  471 

98.  If  the  moon  is  a  sphere  2170  miles  in  diameter,  about  how 
many  million  bushels  would  she  hold  if  hollow  ?  and  how  many- 
yards  of  cloth  a  yard  wide  would  it  take  to  cover  her  ? 

2170  mi.  =  137,491,200  in. 
F=  0.5236  xZ>3. 

log  137,491,200=^  =  24.4149 
log  0.5236=    9.7190-10 

colog         2150.42=    6.6675-10 

log  V=  20.8014 

V=  633,000,000,000,000,000,000  bu.  (1)  Ans. 

2170  mi.  =  3,819,200  yds. 

xog  3,819,2002  =  13.1640 
log        3.1416=    0.4971 

log  >S'=  13.6611 

S=  45,800,000,000,000  yds.  (2)  Ans. 

99.  If  the  earth  is  7920  miles  in  diameter,  and  the  air  is  40  miles 
deep,  how  many  cubic  miles  of  air  are  there  about  the  planet  ' 

7920  +  80  =  8000  log    7920^  =  11.6961 

log   80003  =  11.7093  log  0.5236=    9.7190-10 

log  0.5236  =    9.7190  -  10  UAlbl 

11.4283  =  log  260,100,000,000 

=  log  268,100,000,000.  268,100,000,000 

260,100.000,000 

8,000,000,000  cu.  mi. 
Ans. 


100.  What  is  the  difference  between  2  feet  square  and  2  square 
feet  ?  between  a  foot  square  and  a  square  foot  ?  between  half  a  foot 
square  and  6  in.  square? 


472  ARITHMETIC. 


"  2  feet  square  "  means  a  square  2  ft.  on  a  side  ;  "  2  square  feet," 
any  surface  equivalent  in  area  to  two  squares  each  1  foot  on  a 
side.  A  "  foot  square  "  is  a  square  ;  while  a  square  foot  is  an 
equivalent  area  in  any  shape.  "Half  a  foot  square"  is  am- 
biguous. Half  "a  foot  square"  is  half  a  square  foot,  while 
"half  a  foot"  square  is  6  inches  square  ;  that  is,  one-fourth  a 
square  foot. 

101.  Find  the  volume  of  a  square  frustum  of  which  the  base  is  3 
ft.  square,  top  2  ft.  square,  and  height  4  ft. 

F=  ^  X  4[>/3^^^r22  +  32  +  22]  =  ^(6  +  9  +  4)  =  25^  cu.  ft.  Ans. 

102.  Find  the  capacity  in  liquid  quarts  of  a  tin  pan  10  in.  in 
diameter  at  top,  8  in.  in  diameter  at  bottom,  and  4  in.  deep. 

Volume  of  whole  cone  =  |  X  0.7854  D'^h  =  ^x  0.7854  X  100  X  20 

=  523.6  cu.  in. 
Volume  of  part  cut  off  =  }  of  0.7854  D'%  =  ^x  0.7854  x  64  X  16 
=  268.08  cu.  in. 
523.6  -  268.08  =  255.52  cu.  in. 
255.52  -  Afi  =  4  42  qts.  Ans. 

103.  How  many  hektoliters  will  a  circular  vat  hold  5°>  in  diam- 
eter at  the  top,  4.57™  at  the  bottom,  and  1.17"  deep? 

Area  of  top    =  5»  x  0.7854  =  19.635o«». 
Area  of  base  =  4.57»  X  0.7854  =  16.4030". 
V19.635  X  16.403  =  17.946. 
1(19.635  +  16.403  +  17.946)  x  1.17  =  21.054«''>»  «  210.54'».  Ans. 

104.  Find  the  area  of  an  ellipse  8  in.  by  11  in.;    of  an  ellipse 
15  in.  by  21  in. 

0.7854  X  11  X    8  =  69.115  sq.  in.  (1)  Ans. 
0.7854  X  15  X  21  -  247.401  sq.  in.  (2)  Ans. 


TEACHERS     EDITION. 


473 


105.    The  ends  of  a  cord  100  ft.  long  are  fastened  to  stakes  placed 

80  ft.  apart  on  level  ground.     A  ring,  to      

which   a   kid  is  tied,  plays   freely   on  the        \      40 
cord.     How  far  from  the  straight  line  join- 
ing the  stakes  can  the  ring  be  pulled?   What 
are  the  diameters  of  the  ellipse  which  the 
kid  can  graze  ?     How  many  square  feet  in  the  ellipse  ? 

The  cord  can  be  pulled  from  AB 

V502  -  402  _  V900  =  30  ft.  (1)  Ans. 
The  diameters  are  100  ft.  and  60  ft.  (2)  Ans. 

0.7854  X  60  X  100  =  4712.4  sq.  ft.  (3)  Ans. 


106.  Using  the  same  rope  as  in  the  last  problem,  but  putting  the 
stakes  25  ft.  apart,  how  many  per  cent  is  the  kid's  pasturage  in- 
creased ? 

The  rope  can  now  be  pulled  from  AB  a,  distance  of 

VSO'^  -  12.5^  =  V2343.75  =  48.4  ft. 
Hence  the  diameters  are  96.8  and  100. 
Area  =  96.8  X  100  x  0.7854  =  7602.7. 
7602.7  -  4712.4 


Hence, 


4712.4 


of  100  =  61.3. 


61.3%.  Ajis. 


107.  A  cylindrical  log,  11  in.  in  diameter,  is  sawed  off  on  such  a 
slant  that  the  pieces  are  8  in.  longer  on  the  longest  than  on  the 
shortest  side.  Find  the  dimensions  of  the  ellipse  thus  made,  and  its 
area. 

The  shorter  diameter  is  evidently  the  diameter  of  the  log,  or  11 


The  longer  diameter  is 


VlP  +  82  =  V'121  +  64  =  V\.6o 
Area  =  13.6  x  11  X  0.7854. 
log     13.6  =  1.1335 
log        11  =  1.0414 


13.6  ft. 


log  0.7854  =  9.8951 
2.0700 


10 


log  117.5  sq.  in.  Ans. 


474  ARITHMETIC. 


108.  Find  the  length  of  a  pendulum  beating  half-seconds ;  of  a 
pendulum  beating  quarter-seconds. 

The  length  of  pendulums  is  inversely  as  the  square  of  the  num- 
ber of  vibrations  in  a  given  time. 

Hence,  a  half-seconds  pendulum  will  be  ^  X  ^  =  }  the  length  of 
a  seconds  pendulum,  or  \  of  39.138  in.  =  9.784  in,  (1)  Ans. 

A  quarter-seconds  pendulum  will  ^e  |  X  ^  =  yV  ^^^  length  of  the 
seconds  pendulum,  or  -^  of  39.138  in.  =  2.446  in.  (2)  Ans. 

109.  How  man)'  centimeters  long  is  a  pendulum  swinging  80  times 
a  minute  ?     A  pendulum  swinging  30  times  a  minute  ? 

The  first  pendulum  must  be  (f^)*  of  39.138  in.  =  22.014  in. 

2.54x22.014    =55.92«'».     (1)  Ans 
The  second  pendulum  must  be  (f^)^  of  39.138  in.  =  156.552  in. 

2.54  X  156.552  =  397.64«'".  (2)  Am. 

110.  If  a  cannon-ball  be  suspended  by  a  fine  wire  176  ft.  long  in 
th(!  central  well  of  the  Bunker  Hill  Monument,  how  many  times 
a  minute  will  it  swing? 

V2112  :  V39l38  :  :  60  :  a;. 
^_60x>/39.138 
V2IT2 

log    60  =  1.7782 

logV39.138  =  0.7963 
cologV2ll2    =8.3377-10 

0.9122  =  log  8.17.       8.17.  ^rw. 

111.  Find  the  lifting- power  of  a  hydraulic  press,  the  plunger 
being  l""  in  diameter  and  driven  with  a  force  of  lOO''*,  if  the  lifting- 
piston  is  1"'  in  diameter. 

Suppose  the  plunger  and  piston  square :  the  one  would  press  on 
a  surface  of  li''"',  the  other  on  a  surface  of  10,000*»<'«.  By 
driving  the  plunger  in  10*"*',  you  force  10*»™  of  water  under 
the  piston.  But  as  this  is  spread  under  10,000'>«'°,  you  raise 
the  piston  thereby  only  0.001«=™;  that  is,  only  0.0001  part  of 
the  way  you  move  the  plunger.  Hence,  by  the  principle  of 
virtual  velocity  (what  is  lost  in  time  is  gained  in  power),  you 
lift  the  piston  with  10,000  times  your  force  of  100*8  applied. 
The  lifting  power,  in  other  words,  is  1,000,000*«,  or  1000*. 


teachers'  edition.  475 

112.  If  the  pluuger  is  |  in.  in  diameter,  and  is  driven  with  a 
force  of  1000  lbs.,  how  much  can  it  lift  with  a  lifting-piston  4  ft.  in 
diameter  ?  • 

i  in.  :  4  ft.  =  1 :  96, 
12  :  962  ==  1  :  9216. 
9216  X  1000  lbs.  =  9,216,000  lbs.  Ans. 

113.  If  the  plunger  is  2  in.  in  diameter,  and  is  driven  with  a  force 
of  1000  lbs.,  how  much  can  it  lift  with  a  lifting-piston  2  ft.  in 
diameter  ? 

2  in.  :  2  ft.  :  :  1  :  12, 
12 :  122  =  1 :  144 
144  X  1000  lbs.  =  144,000  lbs.  Ans. 

114.  The  water  stands  in  a  fissure  in  a  rock  10°^  high  and  12™ 
long.  What  pressure  is  exerted  to  split  the  rock  on  the  lowest 
meter's  width  ?  on  the  highest  meter's  width  ?  in  the  whole  fissure  ? 

1  X  12  X  9.5  =  114«'"^  =  114  t.  (1)  Ans. 

1  X  12  X  0.5  =      e^""*  =      6  t.  (2)  Ans. 

10  X  12  X     5  =  600«^°^  =  600  t.  (3)  Ans. 

115.  A  dam  is  100  ft.  long  and  10  ft.  deep,  and  the  water  is  just 
flowing  over  it.     What  pressure  is  exerted  over  the  lowest  two  feet 

of  the  dam  ? 

2  X  9  X  100  =  1800  cu.  ft. 
1800  X  62^  lbs.  =  112,500  lbs. 
112,500  -^  2000  =  56|  t.  Ans. 

116.  What  velocity  in  meters  a  second  will  a  cannon-ball  acquire 
in  falling  three-quarters  of  a  second  ?  in  falling  three  and  a  quarter 
seconds  ? 

f  of  9.806'°  =  7.3545'«.  (1)  Ans. 
31  of  9.806'°  =  31.869'".  (2)  Ans. 

117.  How  long  will  it  take  a  leaden  ball,  rolling  off  a  table  29  in. 

high,  to  reach  the  floor? 

29  in.  =  0.7366'°. 

4.903  :  0.7366  =  12 :  x\ 


^=a/4: 


0.7366 
903* 


476  ARITHMETIC. 


log  0.7366  =  9.8673 -10 
colog  4.903    =9.3095-10 

9.1768  -  10 
10.0000  -  10 


2)19.1768-20 
0.3876  sec.  Ans.  9.5884  -  10  =  log  0.3876. 


118.   What  velocity  will  a  crowbar  attain  in  falling  endwise  from 
a  balloon  2000'n  high  ?     How  long  will  it  be  in  coming  down  ? 

4.903"' :  2000'"  -  12 :  a;2. 


X 

/2000 
\4.903 

log 

2000 

=  3.3010 

colog 

4.903 

=  9.3095  - 
2)2.6105 

10 

1.3053  = 

log 

20.2. 

20.2  sec.  (2)  Ans. 

20.2  X  9.806"' =1 98.08 12". 
198"°  per  sec.  (1)  Ans. 

119.  What  velocity  will  a  crowbar  attain  in  falling  endwise  from 
a  balloon  one  mile  and  a  quarter  high  ?  How  long  will  it  be  com- 
ing down  ? 

4.903"'  =  16.086  ft. 
1^  mi.  =  6600  ft. 
16.086:  6600  =  'l»:r». 

^600" 


V 


16.086 

log    6600  =  3.8195 
colog  16.086  =  8.7936  -  10 

2)2.6131 
20.26  see.  (2)  Ans.  1.3066  -  log  20.26. 

2  X  20.26  X  16.086  ft.  =  651.80  fl. 

651.8  ft.  per  sec.  (1)  Aita.   . 


teachers'  edition.  477 

120.  If  Carisbrook  Well  is  210  ft.  deep,  how  long  after  a  pebble 
is  dropped  will  it  be  before  it  is  heard  to  strike  the  bottom,  if  its 
velocity  is  reckoned  at  32  ft.  at  the  end  of  a  second,  and  the  velocity 
of  sound  is  1120  ft.  a  second? 

16:210=P^ 
X  =  V-W- 
log  210  =  2.3222 
colog    16  =  8.7959-10 

2)1.1181 
0.5591  =  log  3.623. 
3.623  sec.  in  falling. 

yYa^V  ="  0,188  sec.  sound  in  rising. 
3.623  +  0.188  =  3.811  sec.  Ans. 

121.  On  the  same  suppositions  as  in  Ex.  120,  how  long  after  a 
pebble  is  dropped  will  it  be  heard  to  strike  the  bottom  of  a  ventilating 
shaft  1600  ft.  deep  ? 

16  :  1600  =  12  :  a;2 
X  =  V^ff^  =  VlOO  =  10  sec. 
10  +  1.429  =  11.429  sec.  Ans. 

122.  If  a  pebble  is  dropped  over  a  precipice,  and  is  heard  to  strike 
the  bottom  in  7^  sec,  how  far  has  it  fallen,  on  the  same  suppositions 
asm  Ex.  120? 

Assume  745  ft.  and  750  ft.  to  be  the  distance. 


16  :  745  =  P  :  a;2 

16:750  =  P:a;'» 

x  =  VW 

x=V^~ 

log  745  =  2.8722 

log  750  =  2.8751 

colog   16  =  8.7959-10 

colog   16  =  8.7959-10 

2)1.6681 

2)1.6710 

0.8341 

0.8355 

=  log  6.825. 

=  log  6.847. 

,7^Vtt  =  0.665. 

,7,a,o,  =  0.670. 

6.825  +  0.665  =  7.49  sec. 

6.847 +  0.670  =  7.51 7  sec. 

An  error  of -0.01. 

An  error  of  +  0.017. 

Difference  between  errors  is  0.027. 


478  ARITHMETIC. 


Error  of  745  :  5     =  0.01  :  0.027. 
Error  of  745  :  5      =  10  :  27. 
Error  of  745  X  27  =  50. 
Error  of  745  =  f  ^  =  1.8. 

745  +  1.8  =  746.8  ft.  747  ft.  Ans. 

123.   A  pebble  dropped  down  a  shaft  is  heard  to  strike  the  bottom 
in  3  sec.  after  it  begins  to  fall.     Find  the  depth  of  the  shaft. 
Assume  130  and  140. 


16  :  130  =  12  :  x\ 

16  :  140  =  P  :  a;2 

X  =  V^^^. 

^=vw. 

log  130  =2.1139 

log  140  =  2.1461 

2)0.9098 

2)0.9420 

0.4549 

0.4710 

colog    16  =  8.7959-10 

colog    16  =  8.7959  -  10 

=  log  2.851. 

=  log  2.958. 

t¥2'V  =  0-116  sec. 

1^17  =  0.125  sec. 

5.851 +0.116  =  2.967  sec. 

2.958 +0.125  =  3.083  sec. 

An  error  of  -  0.033. 

An  error  of  +  0.083. 

Difference  between 

errors  is  0.117. 

Error  of  130 : 

10  = 

0.034  :  0.117. 

Error  of  130 : 

10  = 

34:  117. 

Error  of  130 

= 

'\f^     2.9  ft. 

130  +  2.9  =  132.9  ft.            133  ft.  Am. 

124.   How  long  will  it  take  a  ball,  rolling  off  a  table,  to  drop  1«"? 
1  in.?  10<"«?  6  in.? 


4  903m  .  icm  _  12  .  ^7 

4.903  :  0.01  =  l:x\ 

^=^V4.903' 

4.903™  :  1  in.  =  1«  :  x". 

4.903  :  0.0254  =  P  :  x\ 

^        /0.0254 
"~V4.903- 

log  0.01    =8.0000-10 
colog4.903  =  9.3095 -10 

log0.0254  =  8.4048 -10 
colog  4.903   =9.3095-10 

2)17.3095-20 
8.6548-10 
=  log  0.04517. 
0.04517  sec.  (1)  Ana. 

2)17.7143-20 
8.8572  -  10 
=  log  0.07198. 
0.07198  sec.  (2)  Ans. 

teachers'  edition.  479 

4.903°» :  10«™  =  l''^ :  .t2.  4.903"' :  6  in.  =  1^ :  x\ 

4.903  :  0.1  =  P  :  x\  4.903  :  0.1524  =  1^ :  x^. 

\  4.903  \  4.903 

log  0.1      =  9.0000  -  10  log  0.1524  =  9.1830  -  10 

colog  4.903  =  9  3095  -  10  colog  4.903  =  9.3095  -  10 

2)18.3095-20  2)  18.4925  -  20 

9.1548-10  9.2463 -JO 

=  log  0.1428.  =  log  0.1763. 

0.1428  sec.  (3)  Ans.  0.1763  sec.  (4)  Ans. 

125.  With  what  velocity  will  water  flow  through  a  hole  9  ft. 
below  the  surface? 

\/9  :  Vl6  =  3:4.  f  of  32  =  24  ft.  Ans. 

126.  With  what  velocity  will  water  leave  a  fountain  having  free 
play,  and  a  head  of  25  ft.?  a  head  of  100  ft.  ? 

V25  :  VT6  =5:4.  I  of  32  =  40  ft.  (1)  Ans. 

VIOO  :  Vl6  =  10  :  4  =  5  :  2.         f  of  32  =  80  ft.  (2)  Ans. 

127.  If  a  hole  in  the  side  of  a  cistern  4  ft.  below  the  surface  of  the 
water  is  delivering  10  gals,  an  hour,  how  many  gallons  would  it 
deliver  with  5  ft.  more  head  ? 

\/4  :  V9  =  10  :  x.         2  :  3  =  10  :  a;.         x  =  -%^-  =  15  gals.  Ans. 

128.  If  a  pipe  2  in.  in  diameter,  and  1  ft.  long,  inserted  in  a  dam, 
the  head  of  water  being  kept  constant,  delivers  4  gals,  a  minute, 
how  many  gallons  a  minute  may  be  expected  when  another  pipe  of 
the  same  length,  but  2^  in.  in  diameter,  is  substituted  for  the  two- 
inch  pipe? 

22 :  (2^)2  =  4  :  X.  4  :  6^  =  4  :  a;.  x  =  6^  gals.  Ans. 

129.  If  a  one-inch  pipe,  20  in.  long,  is  substituted  for  the  two- 
inch  pipe,  1  ft.  long,  in  Ex.  128,  and  the  flow  is  found  to  be  5  pts.  a 
minute,  what  part  of  the  diminution  is  due  to  the  smaller  area  of  the 
orifice,  and  what  part  to  the  increased  friction  on  the  sides  of  the 
longer  pipe  ? 

22  :  12  =  4  :  a;.      4  :  1  =  4  :  a;,     a;  =  1  gal.     4-1=3  gals.  (1)  Ans. 
1  gal.  =  8  pts.  8-5  =  3  pts.  (2)  Ans. 


480  ARITHMETIC. 


130.  A  miller  is  using  water  flowing  through  the  gate-way  under 
4  ft.  head.  How  much  more  work  could  he  do  if  the  head  was  raised 
to  9  ft.  ?  how  much  more  if  the  head  was  raised  to  25  ft.  ? 

V#:  \/93  =  1  :  X.  Vl^:  VW  =  1  :  X. 

V64  :  V729  =  1 :  a.  V64  :  V 15625  =  1 :  x. 
8:  27  =  1:  a;.  8:  125=1:  a;, 

a;  =  Y  =  3|.  (1)  Ans.  x  =  J-f^  =  15f .  (2)  Ans. 

131.  If  a  top  3  in.  in  diameter  is  making  200  revolutions  a  second, 

with  what  force  does  the  outer  layer  pull  away  from  the  centre  ? 

1.227  X  i  X  200-^  =  6135. 

6135  times  the  weight  of  the  material.  Am. 

132.  If  a  sling  30  in.  long  contains  a  stone,  and  is  whirled  round 
80  times  a  minute,  what  is  the  force  pulling  on  the  string? 

30  in.  =  2\  ft. 
1.227  X  2^  X  {l\f  =  5.453. 
5,453  times  weight  when  even  with  the  hand. 
5.453  —  1  =  4.453  times  weight  when  highest. 
5.453  +  1  =  6.453  times  weight  when  lowest. 

133.  With  what  force  does  a  locomotive  running  at  30  mi.  an 
hour,  on  a  curve  of  800  ft.  radius,  bear  against  the  outer  rail  ? 

30  mi.  per  hr.  =  \  mi.  per  min.  =  y^^  mi.  per  sec.  =  44  ft.  per  sec. 

Diameter  is  1600  ft. 

3.1416  X  1600  =  5026.56  ft.  circumference. 

44 

Hence,  the  locomotive  makes revolutions  per  sec.   . 

5026.56  *^ 


Force  =  1.227  X  800  x 


\^5026.56y 


log  1.227  =  0.0889 
log  800  =  2.9031 
log  44»  =  3.2870 

colog  5026.56'  =»  2.5974  -  10 

log  force       =  8.8764  -  10  =  log  0.07523. 
0.07523  times  weight.  Am. 


TEACHEES'    EDITION.  481 

134.  If  washed  wool  is  put  wet  into  a  wire  basket  1.2'"  in  diam- 
eter, and  the  basket  is  set  to  spinning  at  the  rate  of  180  revolutions 
a  minute,  with  what  force  is  water  wrung  out  of  the  wool  ? 

180  revolutions  per  min.  =  3  revolutions  per  sec. 

4.025x0.6x32  =  21.735. 
21.735  times  its  weight.  Ans. 

135.  If  steel  pens  are  revolved  in  a  basket  32<""  in  diameter,  17 
revolutions  a  second,  with  what  force  is  the  oil  drained  from  them? 

4.025x0.16x172  =  186.116. 
186.116  times  weight.  Ans. 

136.  How  strong  a  horizontal  pull  on  a  chain,  weighing  half  a 
pound  to  the  yard,  is  required  to  make  the  lowest  part  curve  with  an 
18-in.  radius?  with  a  6-ft.  radius? 

18  in.  =  i  yd.  6  ft.  =  2  yds. 

I  yd.  weighs  I  lb.  2  yds.  weigh  1  lb. 

.-.  tension  =  I.  (1)  Ans.  .'.  tension  =  1  lb.  (2)  Ans. 

137.  A  |-in.  rope,  weighing  ^  of  a  pound  to  the  yard,  is  fastened 
at  one  end  to  a  staple,  and  near  the  other  end,  on  the  same  level, 
runs  over  a  pulley,  and  has  a  25-lb.  weight  hung  to  it.  What  is  the 
radius  of  its  curvature  at  the  middle  ? 

The  horizontal  tension  is  25  lbs.,  which  represents  100  yds.  of 
rope.     .•.  radius  =  100  yds.  Ans. 

138.  A  shower  wets  the  rope  of  Ex.  137,  and  increases  its  weight 
40%;  what  does  its  radius  now  become? 

The  weight  of  the  rope  being  |f^  =  1  of  what  it  was,  it  takes 
only  f  of  100  yds.  =  71f  yds.  Ans. 

139.  A  steam-tug,  in  attempting  to  move  a  ship,  straightened  her 
hawser  until  the  radius  of  the  lowest  point  was  1980  ft.  The  rope 
was  wet,  and  weighed  3  j  lbs.  to  the  yard.  With  what  force  was  it 
stretched  ? 

1980  ft.  =  660  yds.,  and  3^  x  660  =  2145  lbs.  tension.   .4??.«. 


482  ARITHMETIC. 


140.   A  chain  31  ft.  long  hangs  between  points  on  a  level,  and  sags 
4  ft.  ?     What  is  the  radius  at  the  lowest  point  ? 

^  _  (^ch.XBag)ach.-sag)  _  (15.5  +  4)(15.5-4) 
2  sag  2x4 

=  ^^""^  ^  ^^-^  =  28.031  ft.  Am. 


141.  The  whole  chain,  in  Ex.  140,  weighs  18  lbs.  What  is  the 
horizontal  tension?  What  is  the  distance  of  the  points  apart? 
What  is  the  slant,  or  batter,  of  the  end  of  the  chain  ? 

T==  weight  of  B.  1  ft.  weighs  ||  lb. 

Tension  -  if  X  28.031  =  16.27  lbs.  (1)  Am. 

From  Prop.  III. 

log  ^  span  ==  log  19.5  +  log  11.5  +  log  (1.29  -  1.0607)  -t-  colog4  +  0.0612. 

1.29  -  1.0607  =  0.2293. 

log     19.5  =  1.2900 

log     11.5  =  1.0607 

log  0.2293  =  9.3604 -10 

colog         4  =  9.3979  -  10 

0.0612 


1.1702  =  log  14.79. 
2x14.79  =  29.58.  (2)  Am. 

Batter  =  ^^  -  1.808.  (3)  Am. 
15.5 

142.  A  chain  weighing  l''^  to  the  meter  is  suspended  from  points 
on  a  level;  the  length  of  chain  is  31",  and  it  sags  1.3™.  Find  all 
the  conditions,  and  find  how  much  it  falls  below  a  level  at  10°°  from 
each  end. 

i  ch.  -  15.5».  sag  =  1.3°. 

Radius  -  gch.-fsa^X^ch.-Bag)  ^  16.8  x  14.2 
2  sag  2.6 

=  91.75™    (1)  Am. 
Tension  -  91. 75''K.  (2)  Am. 


teachers'  edition.  483 

From  Prop.  III. 
log     16.8  =  1.2253 
log     14.2  =  1.1523 
log  0.0730  =  8.8633 -10 
colog       1.3  =  9.8861  -  10 
0.0612 
1.1882  =  log  15.42. 

2  X  15.42  =  30.84  span.  (3)  Ans. 

'     Batter  =  ^^  =  5.92.  (4)  Ans. 
15.5  ^  ^ 

For  drop  with  hypotenuse  of  10<=™ 

6  :  10  :  :  1  :  a;. 

X  =  1.667  depression. 
For  drop  with  batter  of  10, 

5.92  :  10  :  :  1  :  «. 

a;  =1.689. 
1.689  depression.  (5)  Ans. 

143,  A  chain  100"  long,  weighing  14  oz.  to  the  foot,  is  suspended 
from  points  on  a  level  80'*'  apart.  What  is  the  sag,  the  batter  at  the 
ends,  and  the  horizontal  tension? 

Assume  that  the  sag  is  26.5™. 
log  I  span  =  log  (^  ch.  +  sag)  +  log  (^  ch.  —  sag)  +  log  [log(^  ch.  +  sag) 

—  log  (2  ch.  —  sag)]  +  colog  sag  +  0.0612 

=  log  (50  +  26.5)  +  log  (50  -  26.5)  +  log  [log  (50  +  26.5) 

-  log  (50  +  26.5)]  +  colog  26.5  +  0.061 2 

=  log  76.5  +  log  23  5  +  log  (log  76.5  -  log  23.5) 
+  colog  26.5 +  0.0612. 

log     76.5  =  1.8837 
log     23.5=1.3711 
log0.5126  =  9.7098 -10 
colog     26.5  =  8.5768  -  10 

0.0612  ' 

1.6026  =  log  40.05. 

I  span  =  40.05.         An  error  of  +  0.05. 


484  ARITHMETIC. 


Assume  that  the  sag  is  26.6"». 
log  ^  span  =  log  76.6  +  log  23.4  -f  log  (log  76.6  -  log  23.4) 
+  colog  26.6  +  0.0612. 


log     76.6  = 

=  1.8842 

log     23.4  = 

=  1.3692 

log  0.5150: 

=  9.7118 

colog     26.6  = 

=  8.5751 
0.0612 

1.6015  =  log  39.95. 

1  span  = 

:  39.95. 

Error  of  26.5  :  0.1  =  0.05 

:0.1 

An  error  of  - 

-  0.05. 

.-.  error  of  26.5  =  0.05. 

1  sag  =  26.5  +  0.05  =  26.55.  Am. 

R-^ 

(50 

+  26.55)(50  -  2e 
53.1 

1.55)     76.55x23.45     .^.^  ^^ 
53.1 

Batter  = 

R       33.806 
\  ch.         50 

=  0.6761.  Am. 

I"  =  3.28  ft.  1  ft.  weighs  |  lb. 

Tension  =  33.806  x  3.28  x  |  lb.  =  97  lbs.  =  44kK   Am. 

144.  Suppose  the  points  of  suspension  in  Ex.  143  to  remain  un- 
changed, and  the  chain  to  be  shortened  5".  What  does  the  tension 
become  ? 

Assume  that  the  sag  is  22.5.  ^ 

log  \  span  =  log  25  +  log  70  +  log  (log  70  -  log  25) 
+  colog  22.5 +  0.0612. 

log        25  =  1.3979 
log        70  =  1.8451 
log  0.4472  =  9.6505 -10 
colog     22.5  -  8.6478  -  10 
0.0612 

1.6025  =  log  40.04. 
\  span  =  40.04.  An  error  of  +  0.04. 

Assume  that  the  sag  is  22.6. 

log  \  span  =  log  70.1  +  log  24.9  +  log  (log  70  -  log  24.9) 
+  colog  22.6  +  0.0612. 


teachers'  edition.  485 

log     70.1  =  1.8457 
log     24.9  =  1.3962 
log  0.4489  =  9.6527  -  10 
colog     22.6  =  8.6459  -  10 
0.0612 


1.6017  =  log  39.96. 
I  span  =  39.96.  Error  of  22.5  :  0. 1  =  0.04  :  0.08. 

An  error  of  —  0.04.  Hence  error  of  22.5  is  0.05. 

.-.  sag  is  22.5  +  0.05  =  22.55°>. 

^  _  24.95  X  70.05 


45.1 

24.95  X  70.05 

X  3.28087  X  14 

45.1  X  16 

log 

24.95  = 

=  1.3971 

log 

70.05  = 

=  1.8454 

log  3.28087  = 

=  0.5160 

log 

14  = 

=  1.1461 

colog 

45.1  = 

=  8.3458  - 

-10 

colog 

16  = 

=  8.7959  - 

-10 

2.0463  =  log  111.3  lbs. 
r=  111.3  lbs.  Ans. 

145.    How  long  a  rope  is  required  between  points  100  ft.  apart  to 
sag  30  ft.?  20  ft.?  10  ft.? 

Assume  that  the  chain  is  120.9  ft. 
log  \  span  =  log  (60.45  +  30)  +  log  (60.45  -  30)  +  log  [log  (60.45  +  30) 
-  log  (60.45  -  30)]  +  log  30  -h  0.0612 
=  log  90.45  -f-  log  30.45  -h  log  (log  90.45  -  log  30.45) 
+  colog  30  +  0.0612. 

log   90.45  =  1.9564 
log   30.45  =  1.4836 
log  0.4728  =  9.6747 -10 
colog        30  =  8.5229  -  10 
0.0612 

1.6988  =  log  49.99. 
\  span  =  49.99.  An  error  of  -  0.01. 


486  ARITHMETIC. 


Assume  that  the  chain  is  121  ft. 
log  i  span  =  log  (60.5  +  30)  +  log  (60.5  -  30)  +  log  [log  (60.5  +  30) 

-  log  (60.5  -  30)]  +  colog  30  +  0.0612 

=  log  90.5  +  log  30.5  +  log  (log  90.5  -  log  30.5) 
+  colog  30  +  0.0612. 

log     90.5  =  1.9566 
log     30.5  -  1.4843 
log  0.4723  =  9.6741 -10 
colog        30  =  8.5229  -  10 
0.0612 

1.6991  =  log  50.01. 
^  span  =  50.01.  An  error  of  +  0.01. 

Error  <>f  120.9  :  0.1  =  0.01 :  0.02. 
.-.  error  of  120.9  is  0.05. 
.-.  chain  is  120.9  +  0.05  =  120.95  ft.  (1)  Ans. 

Assume  that  the  chain  is  109.9  ft. 
log  I  span  =  log  (54.95  +  20)  +  log  (54.95  -  20)  +  log  [log  (54.95  +  20) 

-  log  (54.95  -  20) J  +  colog  20  +  0.0612 

=  log  74.95  +  log  34.95  +  log  (log  74.95  -  log  34.95) 
+  colog  20 +  0.0612. 

log   74.95  =  1.8748 
log   34.95  =  1.5435 
log  0.3313  =  9.5202 -10 
colog        20  =  8.6990  -  10 
0.0612 

1.6987  =  log  49.97. 
i  span  =  49.97.  An  error  of  -  0.03. 

Assume  that  the  chain  is  110  ft. 
log  i  span  =  log  (55  +  20)  +  log  (55  -  20)  +  log  [log  (55  +  20) 

-  log  (55  -  20)]  +  colog  20  +  0.0612 

=  log  75  + log  35  +  Iog(log75-log35)+colog20  r  0.0612 
log        75  =  1.8751 
log        35  =  1.5441 
log  0.3310  =  9.5198 -10 
colog        20  =  8.6990  -  10 
0.0612 


1.6992 -log 50.02. 


teachers'  edition.  487 


An  error  of  +  0.02. 
Error  of  109.9  :  0.1  =  0.03  :  0.05. 
Error  of  109.9:0.1  =  3.5. 

.-.  error  of  109.9      =  ^  =  0.06. 
5 

/.  chain  is  109.9  +  0.06  =  109.96.  (2)  Am. 

Assume  that  the  cnain  is  102.5  ft. 
log  ^  span  =  log  (51.25  +  10)  +  log  (51.25-  10)  +  log  [log  (51.25  +  10) 

-  log  (51.25  -  10)]  +  colog  10  +  0.0612 

=  log  61.25  +  log  41.25  +  log  (log  61.25  -  log  41.25) 
+  colog  10  +  0.0612. 

log   61.25  =  1.7872 
log   41.25  =  1.6154 
log  0.1718  =  9.2350 -10 
colog        10  =  9.0000  -  10 
0.0612 

i.6988  =  log  49.98. 
I  span  =  49.98.  An  error  of  -  0.02. 

Assume  that  the  chain  is  102.7  ft. 
log  I  span  =  log  (51.35  +  10)  +  log  (51.35  ~  10)  +  log  [log  (51.35  +  10) 

-  log  (51.35  -  10)]  +  colog  10  +  0.0612 

-  log  61.35  +  log  41 .35  +  log  (log  61.35  -  log  41.35) 
+  colog  10  +  0.0612. 

log  61.35  =  1.7879 

log  41.35  =  1.6165 

log  0.1714  =  9.2340 

colog        10  =  9.0000  -  10 

0.0612 


1.6996  =  log  50.07. 
^  span  =  50.07.  An  error  of  +  0.07. 

Error  of  102.5  :  0.2  =  0.02  :  0.09. 
Error  of  102.5  :  0.2  =  2  :  9. 
.-.  error  of  102.5     =  f  of  0.2  =  0.044. 
Hence  the  chain  is  102.5  +  0.044  =  102.544  ft.  (3)  Ans. 


488  ARITHMETIC. 


146.  If  4  cu,  in.  of  iron  weigh  1  lb.  avoirdupois,  what  is  the  weight 
of  1  cu.  in.  in  grains  ?     What  is  the  specific  gravity  of  the  iron? 

1  lb.  =  7000  gr. 

.'.  1  cu.  in.  of  the  iron  weighs  \  of  7000  =  1750  grs.  (1)  Ans. 

(1728  -^  4)  X  16  =  6912  oz.        6912  ^  1000  =  6.912.  (2)  Am. 

147.  If  4  cu.  in.  of  iron  weigh  1  lb.,  what  is  the  diameter  of  a 
6-lb.  ball?  ofa32-lb.  ball? 

F=  4  X  6  =  24  cu.  in.  F-  32  x  4  =  128  cu.  in. 

V  =  0.5236  D^.  V=  0.5236  D^. 

24  =  0.5236  i>».  128  =  0.5236  D^. 


i>=;/: 


2i_.  i>  =  ;/-12L. 

5236  ^0.5236 

log  24  =  1 .3802  log  128  =  2.1072 

colog  0.5236  =  0.2810  colog  0.5236  =  0.2810 

3)  1.6612  3)2.3882 

0.5537  0.7961 

=  log  3.578.  =  log  6.253. 

3.578  in.  (1)  Ans.  6.253  in.  (2)  Ans. 

148.  If  a  block  of  iron  (with  all  its  corners  square)  measures  17.36 
in.  by  8.7  in.  by  1.76  in.,  what  does  it  weigh  at  \  lb.  to  the  cubic 
inch?  and  what  would  be  the  diameter  if  cast  into  a  ball,  if  11% 
is  allowed  for  waste  ? 

\  of  17.36  X  8.7  X  1.76  =  66.45  lbs.  (1)  Ans. 

265.82  -  0. 1 1  of  265.82  =  236.58. 

log  236.58  =  2.3739 

colog0.5236  =  0.2810 

3)2.6549 

0.8850  =.  log  7.673. 

7.673  in.  (2)  Ans. 

149.  Answer  the  same  questions  as  in  the  last  examplo,  for  a 
block  of  which  the  dimensions  are  71.4  in.  by  8J  in.  by  3J  in. 

71.4  x3J=.  238. 
238  X  8f  =  2062.67. 
I  of  2062.67  =  515.67  lbs.  (I)  Aiut. 


teachers'  edition.  489 

log  2062.67  =  3.3145 

log     0.89  =  9.9494  -  10 
colog  0.5236  =  0.2810 

3)3.5449  . 

1.1816  =  log  15.19. 

15.19  in.  (2)  Ans. 

150.    What  is  the  diameter  of  a  cylinder  11  in.  long  that  holds 

2  gais. '!' 

2  gals.  =  462  cu.  in. 
F=i7ri)2xA. 
462  =  0.7854  Z)2x  11. 

/       462  .  /    42 


i>=V   ^^^    =v 

^0.7854x11       ^' 


0.7854 


log  42  =  1.6232 
colog  0.7854  =  0.1049 


2)1.7281 
0.8641  =  log  7.313. 

7.313  in.  Ans. 


151.   What  is  the  diameter  of  a  cylinder   9  in.  long  that  holds 
2  gals.  ? 

462  =  0.7854  i)2x  9. 


^  0.7854 


'2 

log  462  =  2.6646 
colog  0.7854  =  0.1049 
colog  9  =  9.0458  -  10 

2)1.8153 
0.9077  =  log  8.086. 

8.086  in.  Ans. 


490  ARITHMETIC. 


152.   What  is  the  diameter  of  a  cylinder  SO'"*  long  that  holds  It? 
10'  =  lOOOO^"™. 
10000  =  0.7854  i)2x  30. 


^^     /lOOO 

^lO.?^ 


■854  X  3 
log    1000  =  3.0000 
colog  0.7854  =  0.1049 
colog  3  =  9.5229  -  10 

2)2.6278 
1.3139  =  log  20.6.  20.6««».  Am. 

153.  What  is  the  circumference  of  a  globe  if  the  square  centi- 
meters of  its  surface  are  three  times  the  cubic  centimeters  of  its  vol- 
ume? 

F=  0.5236  i)». 
/S'=  3.1416  i)2. 
3.14162)2- 3  X  0.5236 Z)». 
Divide  both  sides  by  3  x  0.5236  D^. 
2  =  1). 
Hence,  circumference  =  2  x  3.1416  =  6.2832<'™.  Ans. 

154.  Find  the  diameter  of  a  circle  of  which  the  number  of  inches 
in  its  circumference  is  equal  to  the  square  feet  of  its  area. 

Area  =  ^ir  D^  sq.  ft. 
Circumference  =  ir  i)  ft.  or  12irZ)  in. 
12irD  =  \irD\ 
Multiply  each  side  by  4. 

48TrI)  =  irD\ 
Divide  each  side  by  tD. 
48  =  i). 
.-.  i>  =  48  ft.  Ans. 

155.  How  many  times  does  a  carriage-wheel  3  ft.  2  in.  in  diame- 
ter turn  in  going  a  mile  on  a  smooth  road  ? 

The  circumference  of  the  wheel  is 

3J  X  3.1416  =  9.9484  ft. 

1  mi.  =  5280  ft. 

5280 

.-.  the  wheel  turns  - — - —  =  530.7  timos.  Ans. 

9.9484 


teachers'  edition.  491 

156.  The  top  of  a  wheel  is  at  each  instant  moving  with  twice  the 
velocity  of  the  carriage,  and  is  moving  in  a  curve  whose  centre,  at 
the  instant,  is  as  far  below  ground  as  the  point  is  above  ground. 
What,  then,  is  the  force  exerted  to  separate  the  mud  from  the  top  of 
a  wheel  3  ft.  2  in.  in  diameter,  when  the  carriage  is  moving  at  the  rate 
of  10  mi.  an  hour  ? 

The  carriage  going  at  10  mi.,  the  top  of  the  wheel  goes  at  20  mi. 
per  hour,  or  29i  ft.  per  second.  The  radius  of  the  curve  is 
2  X  3|  =  6|-  ft.,  the  total  circumference  of  which  would  be 
12f  X  3.1416  =  39.7936  ft.     .-.  the  force  is 

1.227  X  6i  X  l^l'lll^y=  4.224  times  the  weight. 

log        1.227  =  0.0889 
log  6i  =  0.8016 

log        (291)2  _  2.9348 
colog  (39.7936)2  =  6.8004  -  10 

0.6257  =  log  4.224.  Ans. 

157.  A  point  in  the  tire,  as  a  spike-head,  moves,  while  the  wheel 
rolls  over  once,  just  four  times  the  diameter  of  the  wheel.  How  far 
does  the  point  travel  while  the  wheel,  3  ft.  2  in.  in  diameter,  travels 
Imi.? 

In  going  one  mile  the  wheel  turns  530.7  times. 

.-.  the  point  travels  3i  x  4  X  530.7  =  6722.2  ft.  Ans. 

158.  An  oil-can  is  formed  of  two  cylinders  connected  by  a  frustum 
of  a  cone.  The  upper  cylinder,  or  neck,  is  6"^  in  diameter,  and  75™°* 
high ;  the  lower  cylinder  is  13°™  in  diameter,  and  153™™  high ;  the 
total  length  ©f  the  can  is  30"™.     Find  its  capacity  in  liters. 

A  square  shaft  to  enclose  the  neck  would  contain 

6  X  6  X  7.5  =  270««™ ; 

one  to  enclose  the  body  would  contain 

13  X  13  X  15.3  =  2585.7««'«. 
The  frustum  to  enclose  the  remainder  would  contain 


(\/6  X  6  X  13  X  13  -f  169  -f  36)  I  of  7.2  =  679.2««™ 

(270  +  2585.7  +  679.2)  x  0.7854  =  2776'«=™  =  2.776V  Ans. 


492  ARITHMETIC. 


159.  A  common  tunnel  is  formed  of  a  frustum  of  a  cone  terminated 
with  a  cylinder.  .The  height  of  the  frustum  is  14*=™,  and  the  diam- 
eters of  the  two  bases  are  175™°*  and  16^'^  respectively.  The  cylinder 
is  8°™  long.     Find  the  capacity  of  the  tunnel  in  liters. 

IQmm  ^  1.6cm. 

The  cylinder  holds  l.G^  x  8  X  0.7854  =  20.48  x  0.7854««. 
The  frustum  holds  ^(17.5'  +  1.6^  +  17.5  x  1.6)  x  0.7854 

=  1571.78  X  0.7854««''°. 
Both  together  hold  (1571.78  +  20.48)  x  0.7854<'«°' 

=  1592.20  x  0.7854«=™  =  1250«»»  =  1.25i.  Am. 

160.  A  pan  is  in  the  form  of  a  frustum  of  a  cone.  The  interior 
measurement  is  10"™  deep,  12°™  across  the  bottom,  and  23*=™  across  the 
top.     Find  the  capacity  of  tlie  pan  in  liters. 

Y(122  +  232  ^  12  X  23)  X  0.7854  =  ^(144  +  529  +  276)  x  0.7854 

=  -V- of  949x0.7854 
=  3163^  X  0.7854  =  2484.5«'"> 
=  2.48451.  ^^s, 

161.  A  stove-pipe  is  4™  long,  26"™  in  diameter,  and  1™™  thick. 
Find  how  many  square  centimeters  of  sheet-iron  it  has  taken  to 
make  it,  if  the  edges  lap  one  centimeter ;  and  give  the  weight  of  the 
pipe,  if  the  specific  gravity  of  the  sheet-iron  is  7.8. 

4m  _  4()Qcin  .      Imm  _  Q  1cm 

S  =  TBh  =  (3.1416  X  26  +  1)  X  400  =  33,072.64«»<^.  (1)  Ans. 

0.1  X  33,072.64  =  3307.26''"™. 
7.8  X  3307.26  =  25,796.659*  =  25.8*«.  (2)  Ans. 

162.  A  spherical  bomb  is  32*™  in  diameter,  and  the  sides  38™™ 
thick.  The  specific  gravity  of  the  metal  of  which  it  is  made  is  7.2. 
Find  its  weight  and  interior  capacity. 

Inside  diameter  is  32  -  2  x  3.8  =  24.4<»». 
V"  0.5236  B^  =  0.5236  x  24.4». 

log  5236  =  9.7190 -10 

log  24.48  =  4.1622 


3.8812  =  log  7607. 
7607°™  =  7.60?.  (1)  Am. 


teachers'  edition.  493 

F=  0.5236  X  323. 
log       323  _  4  5153 
logO.5236-9.7J90- 10 

4.2343  =  log  17,150. 
17  150ccm  _  17  151 

7.2  X  (17.15- 7.607)  =  68.71''«.  (2)  Ans. 

163.  The  diameters  of  a  lamp-shade  are  25*'™  and  7°™;  its  slant 
height  IS  134™*".     Give  its  curved  surface  in  square  centimeters. 

134mm  ^  13.4cm 

25  I  7 

— ^^^-  =  16,  average  diameter. 
2  ^ 

3.1416  X  16  X  13.4  =  673.6<i<'«'.  Ans. 

164.  A  niche  is  formed  like  a  half-cylinder  surmounted  by  a 
quarter  of  a  sphere.  The  height  of  the  cylinder  is  1.2™,  the  diameter 
0.8™.  Find  the  volume  of  the  niche,  and  the  area  of  its  interior 
surface. 

The  volume  of  half-cylinder  =  |(0.82  x  0.7854  x  1.2)  =  301.59'. 
The  volume  of  quarter-sphere  =  ^(0.8^  x  0.5236)  =  67.021. 
301.59  +  67.02  =  368.6r.  (1)  Ans. 

The  surface  of  cylinder  is  i  X  0.8  X  3.1416  x  1.2  =  1.50801™ ;  of 
quarter-sphere,  ^  X  4  X  0.8^  x  0.7854  =  0.5027<i™ ;  of  floor, 
1(0.82x0.7854)  =  0.251 31™. 

1.5080  +  0.5027  +  0.2513  =  2.2620<i™.  (2)  Ans. 

165.  A  steam-boiler  is  formed  of  a  cylinder  terminated  at  each 
end  by  a  hemispherical  cap  of  the  same  diameter.  The  length  of  the 
cylinder  is  3.4™,  interior  diameter  0.8™.  Find  the  number  of  hekto- 
liters  of  water  required  to  fill  the  boiler  half  full. 

The  volume  of  cylinder  is 

3.4  X  0.82  X  0.7854  =  1.709<'b™  =  17.09". 
The  two  caps  form  a  sphere,  the  volume  of  which  is 

0.83  X  0.5236  =  0.268«b™  =  2.68". 

^(2.68  +  17.09)  =  9.89".  Ans. 


494  ARITHMETIC. 


166.  A  hill  482  ft.  high  is  8  mi.  from  the  shore.  How  many  miles 
out  at  sea  is  it  visible  ? 

I  log  482  =  1.3415 

0.1215 

1.4630  =  log  29.04. 
29.04  -  8  =  21.04  mi.  Ans. 

167.  A  sailor  at  the  topmast  80  ft.  above  the  sea  can  just  see  a 
sailor  at  the  topmast  of  a  similar  ship.  How  many  miles  apart  are 
the  vessels  ? 

^  log  80  =  0.9516 
0.1215 

1.0731  =  log  11.83. 
2  X  11.83  mi.  =  23.66  mi.  Ans. 

168.  A  vessel  approaching  Valparaiso  at  daybreak  just  makes 
out  the  peak  of  Aconcagua,  23,000  ft.  high  and  140  mi.  back  from 
the  coast.  How  far  is  the  vessel  from  Tand  if  the  eye  of  the  observer 
is  30  ft.  above  the  water  ? 


VI  of  30  =  V52:5  =  7.25  mi. 
Vfof  23000  =  V40250  =  200.6  mi. 
200.6  -  140  =  60.6. 
60.6  +  7.25  =  67.85  mi.  Am. 

169.  If  Mount  Washington  is  6240  ft.  high  and  76  mi.  in  an  air-line 
from  Cape  Elizabeth,  how  far  out  from  the  Cape  will  its  peak  be 
visible  in  the  ordinary  state  of  the  atmosphere? 

Vl  of  6240  =  VI0920  =  104.5  mi. 
104.5  mi.  -  76  mi.  =  28.5  mi.  Am. 

170.  How  many  acres  of  water  can  a  man  see,  standing  on  a  raft 
with  his  eyes  just  6  ft.  above  water,  and  no  land  in  sight? 

i  log  6 -0.3891 
0.1215 
0.5106 


1.0212  ) 
log  3.1416  =  0.4971  } 
log      640  =  2.8062  J 

4.3246  -  log  21,110.        21,110  A.  Am. 


teachers'  edition.  495 

171.    How  far  would  a  mountain  30,000  ft.  high  be  visible  ?  one 
of  5000  ft.  high  ?  one  of  1000  ft.  high  ? 

^Iog30,000  =  2.2386 
0.1215 


2.3601  =  log  229.2.  229.2  mi.  (1)  Ans. 


l\og  5,000  =  1.8495 
0.1215 


1 .9710  =  log  93.55.  93.55  mi.  (2)  Ans. 


^log   1,000-1.5000 
0.1215 


1.6215  =  log  41.83.  41.83  mi.  (3)  Ans. 

172.  How  high  must  a  mountain  be  in  order  to  be  visible  at  sea- 
levei  50  mi.  ?  100  mi.  ?  150  mi.  ? 

f  of  502  _  4  of  2,500  =  1,429  ft.  (1)  Ans. 
j  of  1002  =  4  of  10,000  =  5,714  ft.  (2)  Ans. 
\  of  1502  _  4  of  22,500  =  12,860  ft.  (3)  Ans. 

173.  How  far  is  a  mountain  1000°>  high  visible?  2000™  high? 

y/lb  X  1000  =  \/l5000  =  122.5'^'».  (1)  Ans. 
Vl5  X  2000  =  V30000  =  173.2'^«'.  (2)  Ans. 

174.  How  far  can  a  man  see  from  the  shore,  if  he  stands  on  a  bluff 
that  raises  his  eye  ll""  above  the  sea? 

Vl5x  11  =  Vl65  =  12.85J^°^.  ^718. 

175.  If  in  steaming  away  from  a  mountainous  island  a  sailor  esti- 
mates his  distance  at  171^"",  when  the  island  disappears  beneath  the 
wave,  how  high  shall  he  estimate  the  mountains  ? 

2  X  log  171  =  4.4660 
colog  152  ==  8.8239  -  10 

3.2899  =  log  1950.  1950™.  Ans. 


496  ARITHMETIC. 


176.  The  flash  of  a  gun  is  seen  7J  sec.  before  the  report  of  the  gun 
is  heard ;  there  is  no  wind,  and  the  thermometer  is  73°  F.  How  far 
off  was  the  gun  ? 

73-57  =  16. 
16  +  1114  =  1130  ft.  per  sec. 
7i  X  1130  =  8475  ft.  Ans. 

177.  A  meteor  was  seen  to  burst;  the  report  followed  in  4  min. 
17  sec.  What  was  its  distance  if  the  average  temperature  of  the 
intervening  air  was  50°  F.  ? 

4  min.  17  sec.  =  257  sec. 
57  -  50  =  7. 

1114-7  =  1107  ft.  per  sec. 
257  X  1107  ^  5280  =  53.88  mi.  Ans. 

178.  Find  the  distances  of  the  last  two  examples  by  §  461. 

73°  F.  =  22.8  C. 
22.8  -  21  =  1.8. 
1.8  X  0.53  =  0.95. 
7^345  +  0.95)  =  2594.6»  =  2.5946^".  (1)  Ans. 
50°  F.  =  10°  C. 
0.53  X  (21  -  10)  =  5.83. 
257  X  (345  -  5.83)  =  87.166«»  =  87.\G^«^.  (2)  Ans. 

179.  How  long  will  it  take  for  an  explosion  at  the  equator  to  be 
heard  at  the  antipodes  of  the  place,  if  the  circumference  of  the  earth 
at  the  equator  be  reckoned  at  40,000'"*',  and  the  average  temperature 
at  the  equator  at  23°  C.  ? 

0.53x(23-21)  =  1.06». 

345  +  1.06  =  346.06"»  =  0.34606^ 
20,000  -*-  0.34606  =  57,793  sec. 

57,793  sec.  =  16  hrs.  3.2  min.  Ans. 

180.  If  an  explosion  at  the  equator  occurs  at  sunset,  and  Ihe  aver- 
age temperature  east  of  the  spot  is  22°  C,  and  that  to  the  west  24° 
C,  how  far  from  the  antipodes  would  the  sounds  meet? 


teachers'  edition.  497 

The  velocity  at  22°  C.  is  345.53«>  per  sec. 
The  velocity  at  24°  C.  is  346. 59°^  per  sec. 

692. 12°*  per  sec.  is  the  velocity  with 
which  the  two  sounds  are  approaching  each  other. 
692.12  :  346.59  :  :  40,000  :  X. 
^^3_4a59x40000j^^_  g3^ 

692.12 
20,030.63  -  20,000  =  30.63>™.  Ans. 

181.  How  far  off  is  the  lightning  when  the  thunder  follows  in  13 
eec,  the  temperature  being  76°  F.  ? 

76 -57 +  1114  =  1133  ft. 
13  X  1133  -^  5280  =  2.79  mi.  Am. 

182.  How  long  would  it  take  sound  to  go  through  a  whispering- 
tube  3  mi,  long,  temperature  61°  F.  ? 

61 -57 +  1114  =  1118  ft. 

3  mi.  =  3  X  5280  =  15,840  ft. 
15,804  ^  1118  =  14.16  sec.  Ans. 

183.  Sound  travels  in  iron  about  10|  times  as  fast  as  in  air.  How 
long,  then,  after  seeing  the  blow  of  a  sledge-hammer  given  on  the 
other  end  of  an  iron  pipe  1|  mi.  long,  may  I  expect  to  hear  the  sound 
by  the  iron  ;  and  how  long  after,  to  hear  the  sound  through  the  air 
in  the  pipe  ;  thermometer  63°  F.  ? 

63  -  57  +  1114  =  1120  ft.  per  sec. 
li  mi.  =  7920  ft. 
7920  -^  1120  =  7.07  sec.  through  air.  (1)  Am. 
1.01  -h  10.5  =  0.67  sec.  through  iron.  (2)  Ans. 

184.  Two  gunners  fire  at  each  other  simultaneously  from  forte  1^ 
mi.  apart ;  the  wind,  at  70°  F.,  blows  steadily  from  one  fort  to  the 
other,  at  11  mi.  an  hour.  How  soon  will  each  hear  the  report  of  the 
other's  gun  ?  Suppose  one  ball  flies  on  the  average  987  ft.  a  second, 
the  other  818  ft.  a  second ;  when  will  each  receive  the  other's  shot? 

70  -  57  +  1114  =  1127  ft.  per  sec. 

li  mi.  =  7920  ft. 

11  mi.  per  hr.  =  16.1  ft.  per  sec 


498  ARITHMETIC. 


The  velocity  of  the  sound  with  the  wind  will  be 

1127  +  16.1  =  1143.1. 
The  velocity  of  the  sound  against  the  wind  will  be 
1127-16.1  =  1110.9. 
7920  -f- 1143.1  =  6.93  sec.  (let  sound).  Am. 
7920  -^  1110.9  =  7.13  sec.  (2d  sound).  Am. 
7920 -^  987  =  8.02  sec.  (1st  ball).  Am. 
7920  -5-  818  =  9.68  sec.  (2d  ball).  Am. 

186.  Sound  travels  in  water  about  4.26  times  as  fast  as  in  air. 
How  many  seconds  sooner  would  the  sound  of  a  torpedo  exploded 
under  water  2  mi.  off  reach  you  by  water  than  by  air,  at  68°  F.  ? 

68  -  67  +  1114  =  1125  ft.  per  sec. 
2  mi.  =  10,560  ft. 
10,560  -5- 1125  =  9.387  sec.  in  air. 

9.387  -^  4.26  -  2.203  sec.  in  water. 
9.387  -  2.203  =  7.18  sec.  Am. 

186.  Required  the  expense  of  painting  the  walls  and  ceiling  of  a 
room  22  ft.  6  in.  long,  13  ft.  6  in.  wide,  and  10  ft.  high,  at  30  cents  a 
square  yard. 

2(22^  +  13^)  X  10  =  720  sq.  ft.,  walls. 

22^  X  13^  =  303.75  sq.  ft.,  ceiling. 
^(720  -f  303.75)  x  $0.30  -  $34.13.  Am. 

187.  In  what  time  would  a  cistern  be  filled  by  three  pipes  whose 
diameters  are  \  in.,  f  in.,  and  1  in.,  when  the  largest  alone  would 
till  it  in  40  min. ;  the  rates  of  flow  being  proportional  to  the  squares 
of  the  diameters  ? 

The  smallest  alone  would  fill  it  in  (J)"  of  40  min.  =  ItH)  min. 

The    other    alone  would  fill  it  in  {jY  of  40  min.  =  71^  min. 

Hence,  in  1  min.,  the  largest  fills  -^^  of  the  cistern, 
the  smallest  fills  yj^  of  the  cistern, 
the  other  fills  ^|^  of  the  cistern, 
and  all  three  together  fill  f\j  +  j^j^  +  ^  J^  =  ^/^  of  it. 

Hence,  it  will  take  1  -*-  ^^"^  =  22|  min.  Ans. 


teachers'  edition.  499 


188.  How  many  gallons  of  water  are  contained  in  a  length  of 
of  50  yds.  of  a  canal,  if  its  width  at  the  top  is  8  yds.  and  at  the 
bottom  7  yds.,  and  its  depth  5  ft.  ? 

The  average  width  is  7|-  yds.  =  270  in. ;  50  yds.  =  1800  in. ; 
5  ft.  =  60  in. 

Volume  =  1800X60X270  ^  ,26.233.8  gals.  An.. 
231  ^ 

189.  Find  the  weight  supported  by  each  of  two  men,  A  and  B, 
who  carry  a  hundred-weight  suspended  on  a  pole  6  ft.  long,  at  a 
point  2  ft.  3  in.  from  A's  end,  if  the  pole  weighs  16  lbs. 

100  4- 16  =  116  lbs.,  total  weight. 
6  -  2.25  =  3.75  ft.  from  B's  end. 
6  :  3.75  =  100  :  X. 

^^100x^75  =  62.5  lbs. 
6 

62.5  +  8  =  70.5,  A.  (1)  Ans. 

116-70.5  =  45.5,  B.  {2)  Ans. 

190.  A  man  who  rows  4  mi.  an  hour  in  still  water  takes  1  hr.  12 
min.^o  row  the  same  distance  up  a  river.  How  long  will  it  take 
him  to  row  down  again  ? 

In  still  water,  the  man  could  row  4  X  1.2  =  4.8  miles  in  1  hr.  12 
rain.  Hence,  the  stream  bears  him  down  0.8  mi.  in  1.2  hrs., 
or  flows  at  the  rate  of  |  miles  per  hour.  When  he  rows  with 
the  stream  he  will  row  4f  miles  per  hour,  and  row  4  miles  in 
f  hr.  =  51f  min.  Ans. 

191.  How  long  a  ladder  will  be  required  to  reach  a  window  40 
ft.  from  the  ground,  if  the  distance  of  the  foot  of  the  ladder 
from  the  wall  is  one-fifth  of  the  length  of  the  ladder. 

Let  I  be  the  length  of  the  ladder. 

\l\a  the  distance  of  the  foot  from  the  wall. 

The  height  of  the  wall  is  ^Z^  -  ^^Y- 

But  40  is  the  height  of  the  wall. 

Therefore  Jp-ILX=  40. 


500 


ARITHMETIC. 


Square  both  sides,  f  |  ^  =  1600. 

Divide  both  sides  by  f^,    P  =  1666.67. 

.-.  I  =  Vl666.b7  =  40.8  ft.  Arts. 

192.  If  3  oz.  of  gold  15  carats  fine  are  mixed  with  7  oz.  12  carats 
fine,  what  will  be  the  fineness  of  the  compound  ?  What  must  be  the 
fineness  of  11  oz.  so  that,  when  added  to  this  compound,  the  whole 
may  be  14  carats  fine  ? 

3  X  15  =  45 
7  X  12  =  84 

10)129 

12.9  carats.  (1)  Ans. 

14 -12.9  =  1.1  carats. 
1.1  X  10  -  11  carats. 
11  -5- 11  +  14  =  15  carats.  (2)  Ans. 


193.   Find  the  surface  of  each  face  of  a  cube  whose  volume  is  14 
cu.  ft.  705.088  cu.  in. 

14  cu.  ft.  705.088  cu.  in.  =  24,897.088  cu.  in. 

24897.0881  29.2 


8 


3  X  202  =  1200 

3(20x9)=   540 

9»=     81 


16897 


508088 


1821  [_16389 
3  X  290»  =  252300 
3(290x2)=     1740 
22=  4 


508088 


254044 
(29.2)2  =  852.64  sq.  in.  Am. 


194.  What  must  be  the  length  of  spermaceti  candles  |  of  an  inch 
in  diameter,  that  6  of  them  may  weigh  a  pound,  if  the  specific  gravity 
of  spermaceti  is  0.943  ? 


teachers'  edition.  501 


V=  0.7854  X  -  X  A  X  6  cu.  in. 

1728 

1  lb.  is  the  weight  of en.  in.  of  spermaceti. 

^  0.943  X  62.5  ^ 

Hence  0.7854  x  -  X  A  X  6  ^^^^ 


82  0.943  X  62.5 


^  _  1728  X  82 


0.7854  X  72  X  6  X  0.943  x  62.5 


log  1728  =  3.2375 

log  82  =  1.8062 
colog  0.7854  =  0.1049 

colog  72  =  8.3098  -  10 

colog  6  =  9.2218  -  10 

colog  0.943  =  0.0255 

colog  62.5  ==  8.2041  -  10 

0.9098  =  log  8.124. 

8.124  in.  Ans. 

195.  A  cylinder  10  in.  across  and  10  in.  high  contains  0.3927  of  a 
cubic  foot  of  water.  How  many  shot  0.1  in.  in  diameter  must  be  poured 
in  to  raise  the  water  to  the  top  ? 

The  volume  of  the  cylinder  is 

0.785398  X  102  x  10  =  785.398  cu.  in. 
0.3927  cu.  ft.  =  678.5856  cu.  in. 
785.398  -  678.5856  =  106.8124  cu.  in. 

0.13  X  0.5236  =  0.0005236  cu.  in.,  vol.  of  1  shot. 
106.8124  ^  0.0005236  =  203,997.  Ans. 

196.  Determine  the  depth  of  conical-shaped  wine-glasses  2^  in. 
across  the  top,  that  60  of  them  may  hold  a  gallon. 

■^jj  of  231  =  3.85  cu.  in.,  the  capacity  of  a  glass. 
2.5  X  2.5  X  0.7854  =  4.90875,  the  area  of  the  top. 

3  X  _M^  =  2.353  in.  Ans. 

4.90875 


502  ARITHMETIC. 


197.  A  train  approaching  a  station  at  the  rate  of  25  mi.  an  hour 
passes  over  two  signals  placed  half  a  mile  apart.  Find  the  interval 
between  the  times  at  which  their  reports  are  heard  at  the  station, 
sound  travelling  at  the  rate  of  1090.2  ft,  per  second. 

25  mi.  an  hr.  is  ^  mi.  in  1^  mm.  =  72  sec. 

When  the  train  reaches  the  second  signal,  the  sound  of  the  first 
signal  is  72  X  1090.2  -  2640  ft.  ahead  of  it ;  and  this  distance, 
divided  by  1090.2,  is  72  sec.  -  2.4  =  69.6  sec.  A7is. 

198.  The  apparent  intensity  of  light  varies  inversely  as  the  square 
of  its  distance.  Find  the  point  between  two  lamps  50  yds.  apart  at 
which  one  appears  twice  as  bright  as  the  other. 

The  distances  are  in  the  proportion  VT  :  V2  =  1  :  1.4142, 
and  2.4142  :  50  =  1  :  a:. 

x=  20.71  yds.,  smaller  distance.  (1)  Ans. 
50  —  20.71  =  29.29  yds.,  greater  distance.  (2)  Ans. 

199.  How  deep  may  a  round  cistern  4  ft.  across  be  made  so  as  to 
be  lined  with  the  same  lead  as  a  cubical  cistern  4  ft.  each  way? 
Compare  their  capacities. 

4  X  4  X  4  =  64  cu.  ft.  in  cubical  cistern. 
5  X  16  =  80  sq.  ft.  of  lead  to  line  it. 
0.7854  X  16  -  12.5664  sq.  ft.  in  bottom  of  round  cistern. 
80  -  12.5664  =  67.4336. 

^^•^■'^^^    =  5.366.  (1)  Ans. 
4x3.1416 

16  X  0.7854  X  5.366  =  67.43  cu.  ft.  in  round  cistern.  (2)  An.. 

200.  The  material  for  lining  a  cubical  cistern  cost  $10;  find  tiie 
cost  of  material  for  lining  two  similar  cisterns  which  shall  each  hold 
one-half  as  much. 

The  cost  is  proportional  to  ( v^)' :  ( v^)'  =  1» :  0.7937«  =  1 :  0.63. 
1:0.63  =  $10:0;. 

_     $10x0.63 


1 
2x16.30  =  112.60.  Ans. 


=  $6.30. 


TEACHERS     EDITION. 


503 


201.    What  distance  can  be  seen  from  the  top  of  a  mountain  4  mi. 
high? 

4  mi.  =  21,120  ft. 
|log  21,120  =2.1622 
0.1215 


192.2  mi.  Ans. 


2.2837  =  log  192.2. 


202.  If  5  excavators  sink  a  circular  shaft  8  ft.  in  diameter  and 
125  fathoms  deep  in  100  dys.  of  10  hrs.  each,  how  many  nights  of  7 
hrs.  each  will  4  excavators  be  in  sinking  a  shaft  6  ft.  in  diameter  and 
75  fathoms  deep  ;  if  the  difficulty  of  working  by  night  is  one-seventh 
greater  than  by  day,  and  the  hardness  of  the  ground  in  the  smaller 
shaft  is  to  that  in  the  larger  shaft  as  7  is  to  5  ? 

5 
3  ^^ 

100  :  X.        ^X6X6XJ^X10X^X8XTXZP^ 

^X^x;^^xyx7x^ 
=  96f  nights.  Ans. 

203.  Find  the  length  of  a  pendulum  that  oscillates  80  times  a 
minute. 

.  802  :  602  ::  39.138  :  ». 
42  :  32 ::  39.138  :  a;. 
16  :  9  =  39.138  :  x. 


4 

5 

82 

62 

125 

75 

7 

10 

7 

8 

5 

7 

9x39.138 
16 


22.015  in.  Ans. 


204.  A  man  whose  weight  is  160  lbs.,  wishing  to  raise  a  rock, 
leans  with  his  whole  weight  on  a  horizontal  crowbar  5  ft.  long,  which 
is  propped  at  the  distance  of  4  in.  from  the  end  in  contact  with  the 
rock.  Find  what  force  he  exerts  on  the  rock,  and  what  pressure  the 
prop  has  to  sustain,  if  the  weight  of  the  crowbar  is  not  reckoned. 

4  :  56  =  160  :  a;. 
40 
^^^lii^  =  2240  lbs.  {l)Ans. 

2240  +  160  =  2400  lbs.  (2)  Ans. 


504  ARITHMETIC. 


205.  A  child  weighing  56  lbs,  is  at  one  end  of  a  plank  16  ft.  long, 
and  a  child  weighing  72  lbs.  is  at  the  other  end.  Find  the  distance 
of  each  child  from  the  fulcrum  when  the  plank  is  used  for  a  see-saw. 

56  :  72  =  7  :  9.  9  ft.  and  7  ft.  Ans. 

206.  In  a  pair  of  nut-crackers  if  the  nut  be  placed  at  a  distance 
of  1  in.  from  the  hinge,  and  the  hand  presses  at  a  distance  of  8  in. 
from  the  hinge,  find  the  pressure  upon  the  nut  for  every  ounce  of 
pressure  exerted  by  the  hand. 

1 :  8  =  1  oz.  :  what  ?  8  oz.  Ans. 

207.  A  substance  is  weighed  from  both  arms  of  a  false  balance, 
and  its  apparent  weights  are  9  lbs.  and  4  lbs.     Find  the  true  weight. 

4  :  a:  =  a; :  9.  a;^  =  36.  a;  =  6  lbs.  Am. 

208.  A  rope  passes  over  a  single  pulley.  How  much  force  is  re- 
quired to  raise  180  lbs.  attached  to  one  end  of  a  rope  if  1%  of  the 
force  applied  is  required  to  overcome  friction  ? 

-V^  of  180  =  181.82  lbs.  Ans. 

209.  A  cask  of  lamp-black  weighing  160''k  is  attached  to  a  rope 
wound  on  an  axle  19°™  in  diameter ;  at  one  end  of  the  axle  is  a  wheel 
175°™  in  diameter.  With  what  force  must  a  man  pull  down  on  a  rope 
passing  over  the  wheel  to  raise  the  cask  if  1  J%  of  the  pull  is  required 
to  overcome  friction  ? 

175  :  19  =  160^<5 :  X. 
32 
1^2<W  =  17.3714".. 

35 
17.3714  +  0.9825  =  1  7.69"k.  Ans. 

210.  The  circumference  of  the  circle  described  by  the  end  of  the 
power-arm  of  a  screw  is  24  in.,  and  by  one  turn  of  the  screw  the  head 
advances  half  an  inch.     If  the  power  is  3  lbs.,  find  the  weight  lifted. 

i  :  24  =  3  :  a;.  2  X  3  X  24  =  144  lbs. 

Less  than  144  lbs.  Ans. 


teachers'  edition.  505 

211.  In  a  screw  used  to  raise  a  load  of  10  t.  the  power  is  50  lbs., 
acting  by  an  arm  4  ft.  long.  Find  the  distance  between  two  consec- 
utive threads. 

4  ft.  =  48  in.  400:1  =  301.6:0:. 

3.1416  X  2  X  48  =  301.5936.  301.6  -i-  400  =  0.754. 

20,000  :  50  =  301.6  :  x.  Less  than  0.754  in.  Ans. 

212.  What  per  cent  of  water  is  oxygen  ?  what  per  cent  hydrogen  ? 

(2x1) +  16  =  2 +  16  =  18. 
^\of  100=lli%H.  {I)  Ans. 
100-ll|  =  88f%O.  {2)  Am. 

213.  What  per  cent  of  quicklime,  CaO,  is  oxygen  ? 

40  +  16  =  56.  ^  of  100  =  28f  %.  Ans. 

214.  What  per  cent  of  water,  HjO,  in  slacked  lime,  CaO^Hg  ? 

Ca  =  40 

0,  =  32  H2=    2 

H2=    2  0  =  16        ^f  of  100  =  24.32%.  Ans. 

74  18 

215.  What  per  cent  of  pure  marble,  CaCOg,  is  oxygen  ? 

40  +  12  +  48  =  100.  /A  of  100  =  48  %.  Ans. 

216.  What  per  cent  of  gypsum  (plaster  of  Paris),  CaSO^  +  2  HjO, 
is  sulphur  ? 

40  +  32  +  64  +  2(2  +  16)  =  136  +  36  =  172. 
tV^  of  100=1811%.  Ans. 

217.  What  per  cent  of  washing-soda,  NajCOg  +  10  HjO,  is  carbon  ? 

46  +  12  +  48  +  10(2  +  16)  =  106  +  180  =  286. 
^V^ofl00  =  4TVT%-  ^ns. 

218.  In  118  lbs.  of  Glauber  salts,  Na^SO^  +  10  HjO,  how  many 
ounces  of  sulphur? 

46  +  32  +  64  +  10(2  +  16)=  142  +  180  =  322. 
^^r^  of  118  X  16  =  187.6  oz.  Ans. 


50G  ARITHMETIC. 


219    How  many  ounces  of  soda,  NajO  +  HjO,  in  7  lbs.  of  borax, 
NajB^O^  +  10  HjO  ? 

46  +  16  +  18  =  80. 
46  +  44  +  112  +  10(2  +  16)  =  202  +  180  =  382. 
7  lbs.  =  112  oz. 
Tj?y\  X  112  =  23.46  oz.  Ans. 

220.  What  per  cent  of  pure  alcohol,  CjHgO,  is  carbon  ?    "What  per 
cent  of  pure  white  marble,  CaCOg,  is  carbon  ? 

24  +  6  +  16  =  46.  40  +  12  +  48  =  100. 

11  of  100  =  52^% %.  (1)  Ans.         ^^  of  100  =  12%.  (2)  Arts. 

221.  What  per  cent  of  pure  acetic  acid  (the  acid  of  vinegar)  is 
carbon,  the  formula  being  CjH^O.^? 

24+4  +  32  =  60.  f^  of  100  =  40%.  A7is. 

222.  How  much  acetic  acid  can  be  obtained  from  12  lbs.  of  alcohol, 
CjHgO,  if  there  is  no  waste? 

CjH^Oj  =  60  acid.  24  +  6  +  16  =  46  alcohol. 

As  alcohol  contains  ^f  of  0,  and  acid  demands  |§,  12  lbs.  of 
alcohol  will  yield  by  combustion  of  H  and  C 
H  X  M  X  12  lbs.  =  7.82  lbs.  Am. 

223.  How  many  grains  of  carbon  in  1  oz.  avoirdupois  of  oxalic 
acid,  r;.,H20^  +  2H/.)? 

24  +  2  +  64  +  2(2  +  16)  =  90  +  36  =  126. 
1750 

^ofWgr.  =  iip-83igr8.  ^n.. 

21 

224.  How   many   milligrams   of  carbon  .  in   3*  of  tartaric  acid, 

40 

24  +  3+48  =  75.  ^^mi^^eo-,,  Ans. 

3«  -  3ooo»«.        ;r^      1 


teachers'  edition.  507 

225.  How  many  kilograms  of  carbon  in   95''8  of  white  sugar, 

144  +  22  +  176  =  342. 
8         5 
Mx^  =  ^O^s,  Ans. 

X9 

226.  The  formula  of  camphor  is  CjoHjgO.     How  many  grams  of 
carbon  in  14''?  of  camphor  ? 

120  +  16  +  16  =  152. 
15 

^  X  1^  =  210000^,, ^,52.6..  ^n. 
19 

227.  In  20''«  of  oil  of  vitriol,  HjSO^,  how  many  grains  of  sulphur  ? 

2  +  32  +  64  =  98. 

16 

^20000  ^320000  ^g530g,^^^_ 

^^1  49 

49 

228.  What  per  cent  of  oil  of  vitriol  is  water  ?  what  per  cent  sul- 
phuric acid,  SOg  ? 

HjSO^  =  98.  ^f  of  100  =  18.37  %  water. 

H2O     =  18.  100  -  18.37  =  81.63  %  sulphuric  acid. 

229.  In  3.58  of  black  oxide  of  iron,  FeO,  how  many  milligrams  of 
iron? 

3.5«  =  3500«»K.  56  +  16  =  72. 

7 


of  ^  =  24500^27221^,^^^3. 


.9 


230.   Red  iron-rust  consists  of  70%  iron  and  30%  oxygen.     Find 
its  formula. 


508  ARITHMETIC. 


Fe  -  56  and  0  =  16.        16  :  56  =  2  :  7.        30  :  70  =  3  :  7. 
First  seek  multiples  of  16  and  56  in  the  proportion  of  30  to  70 ; 

that  is,  of  3  to  7.     .'.  Fe  :  0  =  2  :  3. 
Formula  =  FcjOj.  Ans. 

231.  The  choking  vapor  of  burning  sulphur  is  equal  parts  sulphur 
and  oxygen.     What  is  its  chemical  formula  ? 

S  =  32.        0  =  16.        Oj  =  32.        .-.  SOj.  Am. 

232.  If  the  composition  of  copperas  is  sulphuric  acid  28.9%,  oxide 
of  iron  25.7%,  water  45.4%,  what  is  its  formula  ? 

Water  being  18,  oxide  of  iron  being  72,  and  sulphuric  acid  80, 
first  seek  multiples  of  72  and  80,  in  the  proportion  of  25.7  to 
28.9 ;  that  is,  of  0.8892  to  1.  But  it  is  evident  that  72  and  80 
are  in  almost  exactly  that  proportion.  This  gives  FeSO^ 
+  water ;  and  the  remaining  question  is  to  find  a  multiple  of 
18  which  is  to  152  as  45.4  is  to  54.6 ;  that  is,  which  is  0.8315 
of  152,  or  126.4.  But  7  X  18  =  126  ;  and  the  addition  of  7  parts 
of  water  gives  as  the  complete  formula,  FeSO^  +  7  HjO.  Am. 

233.  Spirits  of  turpentine  is  11.76%  hydrogen  and  88.24%  carbon. 
Give  a  chemical  formula  which  will  express  that  fact.  Compare  now 
with  camphor,  CjoHigO.  What  per  cent  of  oxygen  combined  with 
spirits  of  turpentine  are  required  to  make  camphor  ? 

Hydrogen  being  1  and  carbon  12,  and  88.24  being  almost  exactly 
7.5  times  11.76,  we  seek  the  smallest  multiple  of  12  that  is  7.5 
times  a  whole  number.  This  is  evidently  5  times  12,  equal 
to  7.5  times  8.     Therefore  the  formula  is  C^ll^.  (1)  Ans. 

Double  it  and  add  0,  and  we  have  CioHigO,  the  formula  of  cam- 
phor ;  containing  16  -i- 136  =  11.76%  of  oxygen  added  to  spirits 
of  turpentine.  (2)  Ans. 

234.  Out  of  the  ten  digits,  0,  1,  2,  3,  4,  5,  6,  7,  8,  9,  how  many 
numbers  each  consisting  of  two  figures  can  be  formed  ? 

The  numbers  actually  formed  are  those  from  10  to  99 ;  90  in  all. 
To  show  how  many  can  be  formed,  consider  that  each  digit 
may  be  followed  by  itself  or  by  any  of  the  other  9,  giving  100 
sequences  of  2  figures ;  but  10  of  these  begin  with  0,  and  are 
to  be  rejected,  leaving  90.  Am. 


teachers'  edition.  509 

235.  How  many  odd  numbers  consisting  of  two  figures  can  be 
formed  with  the  ten  digits  ? 

As  10  is  even  and  99  odd,  one-half  the  90,  viz.  45,  are  odd.  Ans. 

236.  With  the  digits  0,  1,  2,  3,  4,  5,  how  many  numbers  between 
1000  and  4000  can  be  formed  ? 

The  digit  in  the  thousands'  place  must  be  1,  2,  or  3.  Each  of 
these  may  be  followed  in  the  hundreds'  place  by  each  of  the 
6  given  digits,  making  18  diflferent  hundreds.  Each  of  these 
may  be  followed  in  the  tens'  place  by  each  of  the  6,  giving 
108  different  tens ;  each  of  which  may  have  6  digits,  giving 
648  numbers  if  we  include  the  1000.  Ans. 

237.  How  many  dominoes  are  there  in  a  set  numbered  from  double- 
blank  to  double-nine  ? 

By  a  domino  is  meant  two  numbers,  side  by  side.  Double- 
blank  is  00.  The  next  pieces  introducing  1  are  01,  11 ;  those 
introducing  2  are  02,  12,  22.  Every  number  introduced  re- 
quires new  pieces  exceeding  by  one  the  number  introduced. 
Thus,  as  we  have  seen  that  to  go  to  double-2  requires  6,  to  go 
to  double-3  will  require  6  -f  4  =  10  pieces  ;  double-4,  10-1-5 
=  15  pieces.  The  ordinary  set  is  15  +  6  -H  7  =  28  pieces ;  and 
for  double-9's  we  should  require  28  +  8  +  9  -F  10  =  55  pieces. 

Ans. 

238.  How  many  different  amounts  can  be  weighed  with  1-lb., 
2-lb.,  4-lb.,  8-lb.,  and  16-lb.  weights? 

With  a  single  piece,  5  amounts  ;  with  2  pieces,  4-f3-f-2-|-l==10 
amounts  ;  with  3  pieces,  10  amounts  ;  with  4  pieces,  5  amounts  ; 
with  all  5  at  once,  1  amount ;  with  the  5  in  their  various  com- 
binations, therefore,  5  -h  10  -f  10  -f-  5  -f  1  =  31  amounts.  Ans. 


,1  1_»    oouo* 


SolittSooUo. 

r-n/i  Missioa  St. 


